Euclid’s Elements: The first 4 axioms

A modern form of the axioms.

1.

2.

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. 3.

4.

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.

• • •

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

not lie on

l

and for every point P that does 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 )

Girolamo Saccheri (1667-1733)

Only

one

of the following: 1.Hypothesis of right angle (HRA).

2.Hypothesis of Obtuse Angle (HOA) 3.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.

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 Bolyai’s 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.

Adrien-Marie Legendre (1752-1833).

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• 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.

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.