C.F. Gauss' Pr\"azisionsmessungen terrestrischer Dreiecke und seine \"Uberlegungen zur empirischen Fundierung der Geometrie in den 1820er Jahren (C.F. Gauss' high precion measurements of terrestrial triangles and his thoughts on the empirical foundations of geometry in the 1820s)

Abstract

In the historical literature there has been an extended discussion on the question, whether the report of Sartorius von Waltershausen about C. F. Gauss checking the largest triangle of the geodetical measurement campaign in the kingdom of Hannover as a kind of ``test'' for the Euclididean nature of physical space can be taken seriously or not. Among others, it was argued that it even was logically impossible for Gauss to do so in the early 1820s (i.e. in particular, before J. Bolyai's, N.I. Lobachevsy's, or even B. Riemann's works). This article shows, in which sense Gauss's methodology of curvature of surfaces, although logically developed in all clarity only for surfaces embedded in Euclidean 3-space, could very well be used already in the early 1820 to investigate the above mentioned question in the sense of physical geometry. Although we do not have a definitive proof of the respective calculations by Gauss's own hand, the latter's account of the situation to contemporaries (correspondents, friend and students) changed clearly between the early and the late 1820s. That speaks very much in favour of a basic correctness of Sartorius von Waltershausen's report on this topic.