Q:

Spherical geometry as an example of non euclidean geometry

I have recently been learning some hyperbolic geometry and the professor briefly mentioned spherical geometry. From a modern, naive point of view, it seems quite easy to show that spherical geometry is an example of non euclidean geometry.
Clearly, this is not true since it took over a 1000 years to do this. Is the apparent difference in difficulty due to my modern viewpoint or were there technical difficulties like with hyperbolic geometry? 

A:

The key modern insight here is the distinction between axioms and models, in other words between syntax and semantics. Euclidean axioms were once thought to characterize a unique space whatever that may be. The idea that truth of a proposition may be relative to a model is a revolutionary idea that we take for granted today.

