Editing Spherical geometry (section)

==Relation to Euclid's postulates==

If "line" is taken to mean great circle, spherical geometry only obeys two of [[Euclid's postulates | Euclid's five postulates]]: the second postulate ("to produce [extend] 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 ("that between any two points, there is a unique line segment joining them"), there is not a unique shortest route between any two points ([[antipodal point]]s 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 [[parallel postulate|fifth (parallel) postulate]], there is no point through which a line can be drawn that never intersects a given line.<ref>[[Timothy Gowers|Gowers, Timothy]], ''Mathematics: A Very Short Introduction'', Oxford University Press, 2002: pp. 94 and 98.</ref>

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 {{nowrap|180°(1 + 4''f'')}}, 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°.