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
The Earth is a sphere in Euclidean space.
The surface of the earth however, is not.
If you travel along a sheet of paper, and make 4, 90 degree turns, the path you make will be a rectangle since you would be moving in Euclidean space. If you did the same thing on the surface of Earth, your path will not be a rectangle, hence a very simple example of why we need non-euclidean geometry.
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