Relation To Euclid's Postulates
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 logically 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. In fact, 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°.
Read more about this topic: Spherical Geometry
Famous quotes containing the words relation to, relation and/or postulates:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“The problem of the twentieth century is the problem of the color-line—the relation of the darker to the lighter races of men in Asia and Africa, in America and the islands of the sea. It was a phase of this problem that caused the Civil War.”
—W.E.B. (William Edward Burghardt)
“The more reasonable a student was in mathematics, the more unreasonable she was in the affairs of real life, concerning which few trustworthy postulates have yet been ascertained.”
—George Bernard Shaw (1856–1950)