Axiom of Euclide

The axiom of Euclide , axiom of the parallels or fifth postulate of Euclide , due to Euclide (born towards -325, died towards -265 in Alexandria), is a Axiome plan: “By a point external on a line, one can lead one and only one parallel to this line”.

Theorem or axiom?

Euclide itself presented this property like an axiom, its fifth axiom. The geometricians who succeeded to him thought that this property rose logically from the first 4 axioms, that it was obligatorily true.

At the 19th century, with the searchs for Lobatchevsky, Poincaré, Riemann, and Klein, one could find other geometries possible and not-contradictory by preserving the first 4 axioms and by changing the fifth, they are called nonEuclidean Géométrie S.

In certain geometries, the sum of the angles of a triangle is higher than 180° (elliptic Géométrie), in others it is lower than 180° (hyperbolic Géométrie). For example, by modifying the 5th axiom as follows: “By a point external on a line, one can make pass an infinity of straight lines parallel on this line, and all different”, one obtains the hyperbolic Géométrie. With this axiom, the Théorème of Pythagore is false most of the time.

However, even at present this step on various axiomatic constructions of the geometry of the plan completely allowed or is not included/understood as modern attempts at demonstration of the property illustrate it, for example Jacques Camü, Démonstration of the postulate of Euclide and its uselessness , the presses of midday, 2006 (ISBN 2-87867-063-9) . In the Insane arts persons, page 471, (republished in 2001, Paris, Editions of Ashes) (ISBN 2-86742-094-6) , the author André Blavier quotes 13 works published between 1862 and 1932 writings by those which the author calls of the more general term of " quadrateurs" who think of showing the postulate of Euclide.

Statement of origin

The true statement of Euclide is of a more complicated form and more intriguing that which is adopted above; “If a line falling on two lines forms the interior angles on the same side smaller than two rights, these lines, prolonged ad infinitum, will meet side where the angles are smaller than two rights”. In modern language that would give “If a line crosses two other lines by determining two interior angles on the same side whose sum is lower than two right angles, then these two lines are cut in the half-plane where the sum is lower than two right angles”.

Axiomatic of the geometry
Axioms of Hilbert: comparison enters the axioms of Euclide and the axioms of Hilbert
Euclidean Géométrie
not-Euclidean Géométrie

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Axiom of Euclide

Theorem or axiom?

Statement of origin

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