Geometry and “Metaphysics of Space” in Gauss and Riemann

Gauss’s research on the principles of geometry and the axiom of parallels have been the subject of study for long time (e.g. Stackel 1933) which has shed light once and for all on his role in the early history of non-Euclidean geometry. It is therefore unnecessary to go through it all over again; what is more interesting here is to examine the development of Gauss’s ideas from another standpoint which emerges from the first testimony of his reflections on a subject that mathematicians had examined in vain right from antiquity: i.e. the possibility of proving the proposition that Euclid had taken as an axiom and formulated in the following terms: “That, if a straight line falling on two stra…