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?
On
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.
To an ancient geometer (or one from the 18th century), spherical geometry would seem to violate Euclid's second postulate, which says that any finite straight line can be extended indefinitely. It is also a problem that two great arcs intersect in two points, rather than one, which would be natural for straight lines. Thus, it would seem as spherical geometry differ from Euclidean geometry in more than the validity of the parallel postulate.