Why do we need non-euclidean geometry? Don’t Euclidean shapes like sphere and hyperbola suffice?

Question

Why do they have to be geometries of their own kind? For example, instead of talking about lines in spherical geometry, can’t we just imagine circles on a literal sphere in Euclidean space?

In: 4

Answer 1

Because a circle on a sphere in Euclidean space is a circle and is treated as such. It has the properties of a circle as defined by Euclid.

But a circle on a sphere in spherical space is a straight line and has the properties of a straight line as defined by Euclid.