I am wondering about the axiomatic method here and Euclid's 5th often comes up. I don't understand why they said that parallel lines could meet after all in Non-Euclidean Geometry. Surely Euclid did not have this exact definition of a line in mind when he made the axiom, no? I simply don't see how the line we have in Non-Euclidean geometry is what Euclid meant, and therefore why this lead people to reconstruct mathematics (along with many other things in history).
It seems like a just a small terminological conflict that can be resolved by realizing that what we have here in Non-Euclidean geometries are geodesics, not lines.
Some mathematical clarification would be helpful on why exactly what we have in say, hyperbolic space, is considered a line. Why (mathematically) did this shake the groundwork of everything so much? Did we actually for a brief time in history think that it was true of Non-Euclidean spaces, and did anything go seriously wrong in the real world over this?