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In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. z Euclidean Parallel Postulate. We need these statements to determine the nature of our geometry. These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry. I. Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. = [7], At this time it was widely believed that the universe worked according to the principles of Euclidean geometry. In his reply to Gerling, Gauss praised Schweikart and mentioned his own, earlier research into non-Euclidean geometry. For example, the sum of the angles of any triangle is always greater than 180. For planar algebra, non-Euclidean geometry arises in the other cases. In Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p.Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one line parallel to L passing through p.In elliptic geometry, there are no parallel lines at all. In analytic geometry a plane is described with Cartesian coordinates: C = { (x,y): x, y }. However, the properties that distinguish one geometry from others have historically received the most attention. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Other mathematicians have devised simpler forms of this property. hbbd```b``^ Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift. + In Euclidean, the sum of the angles in a triangle is two right angles; in elliptic, the sum is greater than two right angles. , Create a table showing the differences of Euclidean, Elliptic, and Hyperbolic geometry according to the following aspects: Euclidean Elliptic Hyperbolic Version of the Fifth Postulate Given a line and a point not on a line, there is exactly one line through the given point parallel to the given line Through a point P not on a line I, there is no line parallel to I. It was independent of the given line must intersect Gauss 's former student Gerling, other axioms besides parallel! 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