Page "Spherical geometry" Paragraph 10

Spherical geometry obeys two of Euclid's postulates: the second postulate (" to produce a finite straight line continuously in a straight line ") and the fourth postulate (" that all right angles are equal to one another ").

However, it violates the other three: contrary to the first postulate, there is not a unique shortest route between any two points ( antipodal points such as the north and south poles on a spherical globe are counterexamples ); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius ; and contrary to the fifth ( parallel ) postulate, there is no point through which a line can be drawn that never intersects a given line.