Marder, Deegan, and Sharon reply: Many people, including Douglas Robertson, believe that Gauss performed his mountaintop measurement to check whether three-dimensional space itself is Euclidean. Ernst Breitenberger’s careful analysis (our reference 1) persuaded us to the contrary. From 1821 to 1825, Gauss spent many months in the field, mapping Hanover with a heliotrope, an instrument he had invented. The mountain peaks forming the corners of his great triangle were base stations; he measured and plotted 26 smaller triangles between them. The edges of all those triangles were assumed to be great circles, projected down to sea level.
Inconsistencies in the measurements were minimized with a global least-squares adjustment, a method Gauss had invented, and he projected them onto flat surfaces with conformal maps, also his invention. The purpose of the great triangle between Hohehagen, Brocken, and Inselberg was to check the results he had obtained by patching together the 26 smaller triangles. He wanted that check both to ensure the map’s accuracy and to measure deviations of Earth’s shape from a perfect sphere. For the latter task he needed to invent differential geometry. Perhaps amid his phenomenal exhibition of creativity, Gauss also wondered whether space itself is Euclidean, but the great triangle is easily justified without invoking that question, and Breitenberger cites much evidence against it.
