non euclidean geometry essay

lines. Constructive approaches and pedagogy edit The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic. Thales' theorem edit Thales' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. It follows that all straight lines in the new geometry wrap back onto themselves. He wrote The Elements, books of postulates and theorems, which paved the way for modern Geometry.

An indirect strategy was used in the efforts to derive the fifth postulate from the other four. Einstein will no doubt go down in history as a great theoretical physicist. Aristotle expanded on this concept explaining that there were three kinds of loci: straight, circular, and mixed (Coolidge)Then in the third century.C. Much of the, elements states results of what are now called algebra and number theory, explained in geometrical language.

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Euclidean geometry started with practical, direct observations of simple figures in a plane. Klein 's work was based on a notion of distance defined by Cayley in 1859 when he proposed a generalised definition for distance. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started. According to a passage written by Proclus, Euclid probably lived after Ptolemy and the pupils of Plato, but came before Archimedes and Eratosthenes. The transformation was not simply one of size or scope but a manifestation of the cultural shifts which were occurring as the medieval age commenced. So: OA OQ OB Now repeat the construction.