Euclid

1
Euclids Postulates

• Two points determine one and only one straight
  line
• 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

2
Euclids 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

3
Saccheris 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 couldnt both be
  obtuse or both be acute and hence are right
  angles.

4
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

5
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 Saccheris quadrilateral are acute

6
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 Saccheris quadrilateral are obtuse.