Non-Euclidean Maths

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Submitted By emily26
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Describe the work of Gauss, Bolyai and Lobachevsky on non-Euclidean geometry, including mathematical details of some of their results. What impact, if any, did the rise of non-Euclidean geometry have on subsequent developments in mathematics?
Word Count: 1912
Euclidean geometry is the everyday “flat” or parabolic geometry which uses the axioms from Euclid’s book The Elements. Non-Euclidean geometry includes both hyperbolic and elliptical geometry [W5] and is a construction of shapes using a curved surface rather than an n-dimensional Euclidean space. The main difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. There has been much investigation into the first five of Euclid’s postulates; mainly into proving the formulation of the fifth one, the parallel postulate, is totally independent of the previous four. The parallel postulate states “that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” [W1] Many mathematicians have carried out extensive work into proving the parallel postulate and into the development of non-Euclidean geometry and the first to do so were the mathematicians Saccheri and Lambert. Lambert based most of his developments on previous results and conclusions by Saccheri. Saccheri looked at the three possibilities of the sum of the angles in a triangle. He found the first to be <180°, one being =180° and the final being >180°. He showed that the first, that the angles total <180°, was viable but only through the use of a new geometry system, non-Euclidean geometry therefore beginning the investigation into this. Although both Lambert and Saccheri contributed to this field of mathematics, the most famous…...

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