Why did anyone ever doubt that a non-Euclidean geometry was possible?. Euclid's axioms effectively "define" a plane. But Euclid's geometry tells us nothing about solids. Wouldn't solid geometry, especially as it relates to solids that are not formed by the intersection of planes, support a "non-Euclidean" geometry? Shouldn't there be a geometry for every shape that can be represented by an algebraic curve moving through, or rotated through, space?