Euclid's geometrical thesis, "The Elements"(c. 300 B.C.E), proposed five basic postulates of geometry. Of These postulates, all were considered self-evident except for the fifth postulate. Discovery of Non-Euclidean geometry had a profound impact on the development of mathematics in the 19th and 20t...
Euclid's geometrical thesis, "The Elements"(c. 300 B.C.E), proposed five basic postulates of geometry. Of These postulates, all were considered self-evident except for the fifth postulate. Discovery of Non-Euclidean geometry had a profound impact on the development of mathematics in the 19th and 20th centuries. For more than two thousand years the Elements served as a mathematical bible, the foundation of the axiomatic method and the source of deductive knowledge. Euclid's postulates, however, have been based on our(or his) intuition of geometric objects. With the discovery of Non-Euclidean geometry, the Elements were scrutinized and logical omissions were found. Thus, the axiomatic method has been divorced from intuition and formalized, which eventually led to the development of Mathematics and Model Theory. The development steps of Non-Euclidean geometry were described in the following five periods. The first step is the period of Gauss, Bolyai and Lovatschewsky who discovered Non-Euclidean geometry which is called Hyprobolic geometry. The second step is the period of Rieman and Beltrami. In this period, Non-Euclidean geometry was further developed by Rieman who created Rieman geometry(elliptic geometry) which enhanced the general tendency of Non-Euclidean geometry. Also, by using differential geometry, Beltrami proved that Hyprbolic geometry does not have relatively to contradiction. The third step is the period of Cayley and Klein who researched the projective characteristics of Non-Euclidean geometry by using projective geometry. The fourth step is the period of Pasch, Peano, Hilbert and Rusell who built the basis of logic on several axioms and then logically described the total structure of geometry. The fifth step is the period of Einstein, Minkowski and Weyl who advocated the physical realization of Non-Euclidean geometry. Einstein's Theory of General Relativity is based on the idea that material bodies distort space and redefine its geometry. Conclusively, this study includes the general history and the broad basic knowledge of Non-Euclidean geometry. Therefore, it is asserted that this study will not only be able to assist those who have a deep concern about geometry and wish to study it, but also assist students working in the graduate school of education.
Euclid's geometrical thesis, "The Elements"(c. 300 B.C.E), proposed five basic postulates of geometry. Of These postulates, all were considered self-evident except for the fifth postulate. Discovery of Non-Euclidean geometry had a profound impact on the development of mathematics in the 19th and 20th centuries. For more than two thousand years the Elements served as a mathematical bible, the foundation of the axiomatic method and the source of deductive knowledge. Euclid's postulates, however, have been based on our(or his) intuition of geometric objects. With the discovery of Non-Euclidean geometry, the Elements were scrutinized and logical omissions were found. Thus, the axiomatic method has been divorced from intuition and formalized, which eventually led to the development of Mathematics and Model Theory. The development steps of Non-Euclidean geometry were described in the following five periods. The first step is the period of Gauss, Bolyai and Lovatschewsky who discovered Non-Euclidean geometry which is called Hyprobolic geometry. The second step is the period of Rieman and Beltrami. In this period, Non-Euclidean geometry was further developed by Rieman who created Rieman geometry(elliptic geometry) which enhanced the general tendency of Non-Euclidean geometry. Also, by using differential geometry, Beltrami proved that Hyprbolic geometry does not have relatively to contradiction. The third step is the period of Cayley and Klein who researched the projective characteristics of Non-Euclidean geometry by using projective geometry. The fourth step is the period of Pasch, Peano, Hilbert and Rusell who built the basis of logic on several axioms and then logically described the total structure of geometry. The fifth step is the period of Einstein, Minkowski and Weyl who advocated the physical realization of Non-Euclidean geometry. Einstein's Theory of General Relativity is based on the idea that material bodies distort space and redefine its geometry. Conclusively, this study includes the general history and the broad basic knowledge of Non-Euclidean geometry. Therefore, it is asserted that this study will not only be able to assist those who have a deep concern about geometry and wish to study it, but also assist students working in the graduate school of education.
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