Euklidov peti postulat

Euklid je uz Arhimeda i Apolonija bio jedan od najvećih antičkih grčkih matematičara. Najpoznatije Euklidovo djelo su Elementi, nastali oko 300. godine prije Krista. Elementi čine sintezu tad poznate matematike, oni su prvi pokušaj sustavne izgradnje geometrije pomoću deduktivne logičke metode. Euklid je sve svoje tvrdnje u Elementima izveo iz samo pet aksioma i pet postulata. Peti Euklidov postulat, poznat kao postulat o paralelama, najviše je kritiziran od strane drugih matematičara. Naime, zbog njegove složenosti pitalo se može li se on dokazati, odnosno je li on teorem ili stvarno postulat. Tijekom više od 2000 godina matematičari su pokušavali dokazati peti postulat i pritom su otkrivene mnoge njemu ekvivalentne tvrdnje. Konačno, u 19. stoljeću, ruski matematičar Nikolaj Ivanovič Lobačevski i mađarski János Bolyai prvi su izveli posljedice geometrije u kojoj on ne vrijedi, no time nije još bila potvrđena njegova nezavisnost od ostalih postulata. To je potvrdio model hiperbolične geometrije Eugenia Beltramija 1868.Euclid, besides Archimedes and Appolonius, was one of the greatest ancient Greek mathematicians. He is best known for his Elements, dated about 300 BC. Elements are a synthesis of the mathematics of its time and they are the first attempt of developing geometry by strictly deductive logical reasoning. Euclid developed his geometry in Elements from five axioms and five postulates. His fifth postulate, known as the parallel postulate, was criticized by other mathematicians. Namely, because of its complexity, the question of it being a postulate or a theorem arose. In about 2000 years many attempts were made to prove it, which resulted in many equivalent formulations. Finally, in the 19th century, the Russian mathematician Nikolai Ivanovich Lobachevsky and the Hungarian János Bolyai described the consequences of a geometry in which this axiom is not valid. The independence of the parallel postulate was finally confirmed by Eugenio Beltrami’s first model of non-Euclidean (hyperbolic) geometry in 1868