Gauss was too brilliant a man to overlook the implication of this fact. If there was some freedom in the choice of a parallel axiom, then one might choose an axiom different from Euclid’s and build a new kind of geometry. Gauss did just this. He pursued the logical implications of a system of axioms which included the assumption that more than one parallel to a given line passed through a given point, and thus created non-Euclidean geometry.
