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Return to the homepage by clicking on the site logo. In elliptic geometry the lines “curve toward” each other and intersect. If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Other mathematicians have devised simpler forms of this property.
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To draw a straight line from any point to any point. To describe a circle with any centre and distance . That all right angles are equal to one another. All of these early attempts made at trying to formulate non-Euclidean geometry however provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate.
These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge. Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid which he didn’t realize was equivalent to his own postulate. 1298, based on al-Tusi’s later thoughts, which presented another hypothesis equivalent to the parallel postulate. Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry.