Non-Euclidean geometry: any practical use at the times of Gauss? I'm making a historical research on the origins of differential geometry, starting with non-Eculidean geometry introduced by Gauss. Reading different historical accounts, what I don't understand is how far he was 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). So, the question is: at the time of the birth of non-Euclidean differential geometry, did Gauss & co. see a theory in an N-dimensional space where parallel lines can meet at infinity just as a theoretical abstraction, or did they already foresee or have in mind some potential applications of it?
 A: Gauß realized that our physical universe could be a non-Euclidean space. He was aware that the question which geometry our universe really has cannot be answered by mathematical considerations, but only by examining the space in which we live, i.e. by geodetic survey. That our physical space could be non-Euclidean was of course a revolutionary idea which could have brought ridicule upon him by the scholarly world. This may be an explanation what he did not publish anything.
Nevertheless Gauß examined whether a possible curvature of space is measurable. Quotation from Wikipedia:

Geodetic survey
In 1818 Gauss, putting his calculation skills to practical use, carried out a geodetic survey of the Kingdom of Hanover (Gaussian land survey), linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.
In 1828, when studying differences in latitude, Gauss first defined a physical approximation for the figure of the Earth as the surface everywhere perpendicular to the direction of gravity (of which mean sea level makes up a part), later called the geoid.
Non-Euclidean geometries
Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.
Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.
Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years." This unproved statement put a strain on his relationship with Bolyai who thought that Gauss was "stealing" his idea.
Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a biographer of Gauss, argues in Gauss, Titan of Science (1955) that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.

During his geodetic survey Gauß performed precise angle measurements of very large triangles. He combined his measurements over the whole kingdom of Hanover, and taking into account earth's spherical curvature, he was finally able to determine the sum of angles in triangles having side length of some hundred kilometers. The result was: If space is bended, then the curvature is so small that it was not measurable by his instruments. Since Gauß exactly determined possible measurement errors, he was even able to specify the maximal value of curvature.
Let me close by adding images of the 10 Deutsche Mark banknote (which was in use in Germany until 31.12.2001 before the Euro banknotes were introduced):
Front side (portrait of Gauß)

Back side (heliotrope and a section of the triangulation network in the kingdom of Hanover)

To get an idea about the size of triangles, the distance between the islands Neuwerk and Wangerooge (upper left triangle) is 45 km.
