Euclid

1
Euclids Elements The first 4 axioms

• A modern form of the axioms.
• For every point P and every point Q not equal to
  P there exists a unique line that passes through
  P and Q.
• For every segment AB and for every segment CD
  there exists a unique point E such that B is
  between A and E and segment CD is congruent to
  segment BE.
• For every point O and every point A not equal to
  O there exists a circle with center O and radius
  OA.
• All right angles are congruent to each other.

2
Equivalent versions of the Parallel Postulate

• That, if a straight line falling on two straight
  lines make the interior angle on the same side
  less than two right angles, the two straight
  lines, if produced indefinitely, meet on that
  side on which are the angles less than two right
  angles. (Euclid ca. 300BC)
• For every line l and for every point  P that does
  not lie on l there exists a unique line m through
  P that is parallel to l. (Playfair 1748-1819)
• The sum of the interior angles in a triangle is
  equal to two right angles. (Legendre 1752-1833 )

3
Girolamo Saccheri (1667-1733)

• Only one of the following
• Hypothesis of right angle (HRA).
• Hypothesis of Obtuse Angle (HOA)
• Hypothesis of Acute Angle (HAA)
• ? way to study the parallel postulate.
• HRA is equivalent to the parallel postulate.
• Axioms 1-4 imply HOA not possible.

4
Adrien-Marie Legendre (1752-1833).

• In 1794 Legendre published Eléments de géométrie
  which was the leading elementary text on the
  topic for around 100 years.
• Always believed that the parallel postulate can
  be deduced from the first four axioms, even after
  seeing Bolyais proof. In 1832 (the year Bolyai
  published his work on non-euclidean geometry)
  Legendre confirmed his absolute belief in
  Euclidean space when he wrote-
• It is nevertheless certain that the theorem on
  the sum of the three angles of the triangle
  should be considered one of those fundamental
  truths that are impossible to contest and that
  are an enduring example of mathematical
  certitude.

5
Adrien-Marie Legendre (1752-1833).

• 1770 defended his thesis in mathematics and
  physics at the Collège Mazarin
• 1775 to 1780 he taught at École Militaire
• 1782 won prize offered by the Berlin Academy for
  treatise on projectiles
• 1783 appointed as adjoint the Académie des
  Sciences, 1791 member.
• Reappointed after Napoleon
• Quarreled with Gauss over priority of reciprocity
  and the method of least squares.
• Worked in number theory, elliptic functions,
  geometry, astronomy,
• Refused to vote for the government candidate,
  lost his pension and died in poverty.

6
Adrien-Marie Legendre (1752-1833).

• Gives correct refutation of HOA.
• Gives attempts of proofs for refutation of HAA.
• His attempt to show that the first four axioms
  imply the parallel postulate leads to yet another
  equivalent version of the parallel postulate
• Through any point in the interior of an angle it
  is always possible to draw a line which meets
  both sides of the angle.