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...Massachusettsinstituteoftechnology physics department the early universe october prof alan guth lecture notes introductiontonon euclideanspaces introduction history of non euclidean geometry is a fascinating subject which described very well in introductory chapter gravitation and cosmology principles applications general theory relativity by steven weinberg here i would like to summarize important points although historical its organization this section describes some essential mathematics should be read carefully euclid showed his elements how could deduced from few denitions axioms postulates one s assumptions however seemed generations mathematicians somewhat less obvious than others assumption known as fth postulate was stated follows if straight line falling on two lines makes interior angles same side right produced indenitely meet that are figure manymathematiciansattemptedtoprovethispostulatefromtheotherassumptions but all these attempts ended failure it discovered replaced an...