Hence the hyperbolic paraboloid is a conoid . The letter was forwarded to Gauss in 1819 by Gauss's former student Gerling. To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. while only two lines are postulated, it is easily shown that there must be an infinite number of such lines. Through a point not on a line there is exactly one line parallel to the given line. 3. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry parabolic, a term that generally fell out of use[15]). His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Then. Yes, the example in the Veblen's paper gives a model of ordered geometry (only axioms of incidence and order) with the elliptic parallel property. Working in this kind of geometry has some non-intuitive results. "Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry, whose importance was completely recognized only in the nineteenth century. When ε2 = 0, then z is a dual number. 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. Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line." The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model, which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. In Euclidian geometry the Parallel Postulate holds that given a parallel line as a reference there is one parallel line through any given point. and this quantity is the square of the Euclidean distance between z and the origin. So circles on the sphere are straight lines . The method has become called the Cayley–Klein metric because Felix Klein exploited it to describe the non-Euclidean geometries in articles[14] in 1871 and 1873 and later in book form. 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