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The solution to the problem of geodesic lines on a biaxial ellipsoid (when two of the axes are equal) is not very hard and can be solved by mathematical tools that existed prior to Gauss - i.e via Clairaut's relation for geodesics on bodies of revolution. However, the solution for triaxial ellipsoids is very difficult, and was made by Jacobi in a remarkable paper fron 1839. My question is indeed intended to learn of Gauss's work in this field - i found, especially in volume 9 of his collected work (about geodesy) many unpublished titles bearing the subject "geodesic lines on ellipsoid" (for example, ERDELLIPSOID AND GEODETIC LINE, on p.65 - 105 in volume 9). I also read that in his treatise on higher geodesy (published at years 1843,1846), he dealt much with calculation with "lines" on ellipsoid.

So, were these calculations just about ellipsoid of revolution (biaxial ellipsoid)? or there was also work about lines in a triaxial ellipsoids (in his published or unpublished work)?

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I do not read German, but I can tell what I know about this. Gauss' worked on a generic ellipsoid (with three different axes) but this work was motivated by geodesy (rather than pure mathematics). So he considered an ellipsoid with three different axes which is close to the sphere (the ratios of the axes are close to $1$), and derived approximate (asymptotic) formulas for small values of parameter responsible for excentricity. His formulas were to be used in geodesy, and their accuracy was higher than that required in geodesy.

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  • $\begingroup$ I voted your answer. Can you please give a more detailed explanation of what you know Gauss did? did he give closed form solutions for geodesics on triaxial ellipsoids or only series expansions? i ask because what i saw gave me the impression that Gauss mainly worked with finite approximations (the first few terms in certain infinite series) and not, for example, elliptic integrals. $\endgroup$
    – user2554
    Apr 18, 2018 at 13:54
  • $\begingroup$ @user2554: as I wrote, I do not read German, so I have not read Gauss' original papers. My knowledge is based on the books of M. Berger, A panoramic view of Riemannian geometry, p. 123-125, and Geometry Revealed, p. 747. Berger refers to the paper of Peter Dombrowski, 150 years after Gauss "Disquistiones generales...", Soc. Math. France, 1979, which I have not read. $\endgroup$
    – Alexandre Eremenko
    Apr 18, 2018 at 16:46
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    $\begingroup$ According to Berger, Gauss did not have a closed form solution for triaxial ellipsoid (this problem solved by Jacobi was proposed by Weierstrass, after Gauss death). You are right: Gauss worked with asymptotic formulas, approximations, as I wrote in my answer). $\endgroup$
    – Alexandre Eremenko
    Apr 18, 2018 at 16:50

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