Question about the personal development of Gauss's understanding of non-Euclidean geometry

Question

I ask this question in order to gain deeper understanding about the gradual process that led Gauss to the realization that non-euclidean geometry is consistent, as well as the method by which he arrived at the laws of "transcendent trigonometry".

The context is the following: in the first years of Gauss's meditations on the parallel axiom (1795-1813), Gauss was very hesitant towards the alternative geometry. Leaving aside the philosophical consequences of the possibility of such a geometry, the methods he tried were mainly geometric constructions in the "euclidean style" but without assuming the parallel axiom. On p.28 of Paul Stackel's essay "Gauss as a geometer", he says the following:

In the period between 1799 and 1804, however, Gauss had tried to make progress in a different way. Notes from 1803 (W. Xi, p. 451) give several attempts to derive the relationships between the sides of a triangle by means of geometric constructions and functional equations derived from them, i.e. by the same method that Gauss also used for the triangle content (see p. 45). At that time his efforts were in vain; perhaps this is the reason why, in contrast to the doubts he expressed about the truth of geometry in 1799, in 1804 he spoke of the hope of finding a way to prove the parallel axiom before his death.

Looking at p.451 of volume 10-1, I saw that Gauss wrote down a multitude of functional equations, and Stackel comments on these notes:

The notebook called Scheda Af contains notes of various kinds, which, as the dates show, date from the years 1801 to 1803. Since the notes printed here are almost at the end of the notebook, it can be assumed that they were written in 1803. Along with the entry No. 99 of the diary - In principiis geometriae egregios progressus feeimus, Br[unovici 1799] Sept., and the statements in the letter to Wolfgang Bolyai of le, Dec. 1799, (Works VIII, p. 159), the above notes form the oldest evidence of GAUSS's preoccupation with the foundations of geometry; they are all the more valuable because they are attempts to gain access to the transcendental trigonometry that Gauss told his student Wächter about during his visit to the Apennine in 1816 (Works VIII, p. 176).

Later - in several letters written by Gauss in the years 1813-1819 - he stated that he developed the alternative geometry to such an extent that he could solve any problem once its characteristic constant is given. This suggests that he probably made advance on the path of deriving relations between sides and angles by the method of functional equations derived from geometric constructions, this time with success.

There are no surviving documents from this period, but much later - between 1840 and 1846 and after his acquaintance with Lobachevsky's works - he did write down such a procedure (in p.255 of volume 8), although these notes might also reflect later understanding of such theorems (according to Stackel). P.22-24 of the article "What did Gauss read in the Appendix?" refers to these notes, explains some of the functional equations written there and comments on their importance. I mentioned this article as a useful source, and maybe a more comprehensive reading of it will lead me to a deeper understanding of this topic, but right now I am busy with other things (I am posting this question as a reminder for myself to return to this topic). For the benefit of the interested reader, I must also mention another Gaussian fragment called "Theorem from Spherology", which is more relevant to spherical geometry, but its methodology is apparently similar.

My main aim is to gain some understanding of these derivations; more specifically, I would like to understand why Gauss's procedure failed in 1803 but succeeded around 1813? What is the essential difference between these two fragments?

(I think this is an often overlooked question - many sources state that Gauss developed non-euclidean (hyperbolic) geometry and than tell the "usual story" of him not intending to publish anything on it - but the exact mathematical nature of his development is seldom spelled out (some of the reasons for that are understandable - Gaussian documents on hyperbolic geometry are scant, and there is no systematic treatise on it from him - not even in his nachlass). Except Stackel's comments, the only article I found that discusses Gauss's method of functional equations is the previously mentioned article.)

Answer 1

Numerous commentators suggest that the shift occured in the period after 1813 until 1816. Do you have a particular primary source of 1813 in mind? Or can you be more concrete in what sense Gauss succeeded in 1813? Regardless here is some secondary discussion hopefully of interest, which will help understand why commentators generally think of the conceptual shift from parallel postulate to non-euclidean geometry happening after 1813 and manifesting by 1816.

First to note that the question is difficult. There is a range of commentary/secondary sources available, though as you rightly point out often not explicitly mathematical. This isn't necessarily a fault of sources but of the meager pickings of primary material. It seems to me that the reason for this is well captured by Gray (2006) who opens:

The claim, made on Gauss's behalf, that he was a, or even the, discoverer of non-Euclidean geometry is very hard to decide because the evidence is so slight.

• Gray, J. (2006). Gauss and non-Euclidean geometry. In Non-Euclidean geometries: János Bolyai memorial volume (pp. 61-80). Springer.

Gray's essay is worth pursuing because he looks to trace the argument, which I won't repeat here. Other texts of interests are Reichardt's Gauss und die nicht-euklidische Geometrie (1976) in particular section 2 will be helpful.

Another source of interest is Halsted (1900) Gauss and the Non-Euclidean Geometry which argues "Thus in 1813 there still is no light." a conclusion certainly compatible with one found in Gray and other commentors. Gray locates a shift between 1813 and 1816:

Evidently he did not then* feel confident in a non-Euclidean geometry. By April 1816 he had shifted his opinion.

(* referring to 1813, my footnote).

This observation of a shift can also be found in Bonola (1912) see pp. 64-75, see specifically p. 66, also observe some mathematical explanations that follow.

• Bonola, R. (1912). Non-Euclidean geometry: A critical and historical study of its development. Courier.

What changed between 1813 and 1816? One theory is articulated in Dombrowski (p. 126):

• P. Dombrowski (1979) 150 Years After Gauss’ «Disquisitiones generales circa superficies curvas», Asterisque 62, Soc. Math. de France, 1979

His work on surveying led him (between 1812 and 1816) to the study of geodesies on ellipsoids of revolution and to the question of determining "all" conformal charts for general curved spaces.

In particular in 1816 Gauss will propose a competition problem that asks maps between planes and spheres (see Dombrowski p. 127).

Further secondary sources of potential interest: Mittler (2005).