The "discovery" of non-Euclidean geometries had profound repercussions on the sciences and philosophy alike and opened a heated debate on the nature of. Lobačevskij: l'invenzione delle geometrie non euclidee. Responsibility: Renato Betti. Imprint: Milano: B. Mondadori, c Physical description: p. ; 22 cm. Geometrie non euclidee. Jump to: navigation, search. Disco di Poincaré - Spiegazioni by Luca Frangella & Girolamo Demasi. Retrieved from.
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Background[ edit geometrie non euclidee Euclidean geometrynamed after the Greek mathematician Euclidincludes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work Elements was written.
In the Elements, Euclid began with a limited number of assumptions 23 definitions, five common notions, and five geometrie non euclidee and sought to prove all the other results propositions in the work.
The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate," or simply the " parallel postulate ", which in Euclid's original formulation is: If geometrie non euclidee straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more geometrie non euclidee than Euclid's other postulates: To draw a straight line from any point to any point.
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To produce [extend] a finite straight line continuously in a straight line. To describe a circle with any centre and distance [radius]. That all right angles are equal to geometrie non euclidee another.
For at least a thousand years, geometers were troubled by the disparate complexity of the fifth geometrie non euclidee, and believed it could be proved as a theorem from the other four.
The theorems of Ibn al-Haytham, Geometrie non euclidee and al-Tusi on quadrilateralsincluding the Lambert quadrilateral and Saccheri quadrilateralwere "the first few theorems of the hyperbolic and the elliptic geometries.
These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including WiteloLevi ben GersonAlfonsoJohn Wallis and Saccheri. These early attempts did, however, provide some early properties of geometrie non euclidee hyperbolic and elliptic geometries.
Geometrie non euclidee - GeoGebraWiki
Khayyam, for example, tried to derive it from an equivalent postulate he formulated from geometrie non euclidee principles of the Philosopher" Aristotle: Another example is al-Tusi's son, Sadr al-Din sometimes known as "Pseudo-Tusi"who wrote a book on the subject ingeometrie non euclidee on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate.
In a work titled Euclides ab Omni Naevo Vindicatus Euclid Freed from All Flawspublished inSaccheri quickly discarded elliptic geometry as a possibility some others of Euclid's axioms must be modified for elliptic geometry to work and set to work proving a great number of results in hyperbolic geometry.
He finally reached a point where he believed that his results demonstrated the impossibility of geometrie non euclidee geometry. His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present.
In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.
In Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. geometrie non euclidee
Le geometrie non euclidee
He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles can be considered half of a Saccheri quadrilateral. He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri geometrie non euclidee Khayyam, and geometrie non euclidee proceeded to prove many theorems under the assumption of an acute angle.
Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius.
He did not carry this idea any further.
CircaCarl Friedrich Gauss and independently aroundthe German professor of law Ferdinand Karl Schweikart  had the germinal ideas of non-Euclidean geometry worked out, but neither published any geometrie non euclidee.
Schweikart's nephew Franz Taurinus did publish important results of hyperbolic trigonometry in two papers in andyet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry. Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry.
Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before,  though he did not publish.
Bolyai geometrie non euclidee his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical geometrie non euclidee.
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