Question

What are the postulates that define Spherical Geometry?

Answer 1

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.

A statement that is equivalent to the parallel postulate is that there exists a triangle whose angles add up to 180°. Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°( 1 + 4f ), where f is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value of f, this exceeds 180°.