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**Ackbeet** So, let's take great circles as straight lines in the spherical geometry. How would you know if a line was a great circle or not? Well, let's assume that you complete the entire line, so it goes all the way around. I'm thinking physically here. Put the sphere on a flat surface such that a point on the line touches the flat surface. Then, if you were to come down with another flat surface parallel to the first flat surface, it should touch the sphere at the opposite point on the great circle. If, instead, you had a small circle (not a great circle), then two points on the opposite sides of the sphere cannot both be on the small circle.