Euclid’s Postulates1. Two points determine one and only one

straight line

2. A straight line extends indefinitely far in either direction

3. A circle may be drawn with any given center and any given radius

4. All right angles are equal

5. Given a line k and a point P not on the line, there exists one and only one line m through P that is parallel to k

Euclid’s Fifth Postulate (parallel postulate)

• If two lines are such that a third line intersects them so that the sum of the two interior angles is less than two right angles, then the two lines will eventually intersect

Saccheri’s Quadrilateral

He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles.

Non-Euclidean Geometry• The first four postulates are much simpler than

the fifth, and for many years it was thought that the fifth could be derived from the first four

• It was finally proven that the fifth postulate is an axiom and is consistent with the first four, but NOT necessary (took more than 2000 years!)

• Saccheri (1667-1733) made the most dedicated attempt with his quadrilateral

• Any geometry in which the fifth postulate is changed is a non-Euclidean geometry

Lobachevskian (Hyperbolic) Geometry

• 5th: Through a point P off the line k, at least two different lines can be drawn parallel to k

• Lines have infinite length

• Angles in Saccheri’s quadrilateral are acute

Riemannian (Spherical) Geometry

• 5th: Through a point P off a line k, no line can be drawn that is parallel to k.

• Lines have finite length.• Angles in Saccheri’s quadrilateral are obtuse.