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Euclid’s Postulates
1. 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)
• 5th: Through a point P
off the line k, at least
two different lines can
be drawn parallel to k
• Lines have infinite
• Angles in Saccheri’s
quadrilateral are acute
Riemannian (Spherical)
• 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.