Gauss on the mountaintops

Michael Marder, Robert Deegan, and Eran Sharon discuss Carl Friedrich Gauss’s measurement of the angles of a triangle across mountaintops (Physics Today, February 2007, page 33 ), but their discussion is simply wrong. They write, “Some believe that Gauss performed his mountaintop measurement to check whether three-dimensional space itself is Euclidean, but in the paper he published at the time he did the work he made no reference to any such question.”

The theoretical spherical triangles of Gauss and Adrien-Marie Legendre that are discussed in the article have edges that are great circles on the surface of the sphere. They thus obey Euclid’s definition of being the “shortest distances” on that surface. But for Gauss’s mountaintop measurements, there is no simple way to construct or observe such great circles, because the light rays that were used to make the angle measurements are not constrained to follow Earth’s assumedly spherical surface. They are, rather, geodesics within 3D space, if one ignores atmospheric refraction. Gauss’s measurement therefore could not have been anything but a test of the Euclidean nature of 3D space. Gauss understood that perfectly well, and his lack of reference to any such test probably simply reflects his well-known reluctance to discuss non-Euclidean geometries.