![]() ![]() The sum of the angles in a triangle is 180 degrees. The unsuccessful efforts to prove the 5th postulate of Euclid showed that many results are equivalent to it. In modern terms, Euclid's 5th postulate is normally stated as: Given a line L and a point P not on the line, there is precisely one line through P in the plane determined by L and P that does not intersect L. As it turned out in this case, failure can be as productive as success. Others tried to prove the postulate by assuming its negation in the expectation that they would produce a logical contradiction. So people set about trying to prove that it followed from the first four axioms. Being more complex and less obvious than the others, it seemed more like a theorem than an axiom. Certain mathematicians had their suspicions about the 5th Euclidean postulate, the one about parallel lines. It was regarded as being not just the only geometry but also as being synonymous with truth. What we now call "Euclidean geometry" was simply called "geometry". His geometry was deeply entrenched in the mind-set of all subsequent mathematicians. This structure endured undisturbed for some two thousand years. ![]() Working circa 300 BC, the Greek mathematician, Euclid, made geometry into an exact discipline by defining five axioms and then building the edifice of geometry on top of these. Non-Euclidean Geometry Non-Euclidean GeometryĪ triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ![]()
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