I题目：浅谈第五公设的产生及其对数学的影响Title：Ontheproductionofthefifthpostulateanditsinfluenceonmathematics摘要欧几里得第五公设，内容是同平面内的一条直线和另外两条直线相互相交，若在某一侧两个内角，和小于两个直角，那么这两条直线，经过无限延长后能在这一侧开始相交，但是由于第五公设内容和叙述都有一定难度，后人对此进行研究和探讨，引发了众多的争论，衍生出来和欧几里得几何不同的新命题，从而导致非欧几里得几何的产生，欧几里得依靠形式思维,通过图形佐证形成体系,在第29个命题中首先运用第五公设，由于第五公设的矛盾性,包括与其他公理、定义、公设的独立性,从古至今的数学家们积极探索,希望能弥补并且拓展新领域,推动何学的发展，而第五公设对几何学发展的最大意义是来自于对第五公设的否定，即非欧几何的诞生.关键词：数学,第五公设,非欧几何,几何学IIAbstractEuclid'sfifthpostulateistheintersectionofastraightlineandtwootherstraightlinesintheplane.Ifthesumoftwoinneranglesononesideoftheplaneislessthantworightangles,thenthetwostraightlineswillintersectonthissideafteraninfiniteextension.DuetothecomplexityofthecontentandnarrativeoftheFifthPost,thecontinuousresearchanddiscussiononitbylatergenerationshasarousedmanycontroversies.Formorethan2,000years,manyscholarshavetriedunsuccessfullytousetherestofthepostulatesandinferencesintheEuclid'sElements,buthaveobtainedsomepropositionsequivalenttothefifthpostulate.Later,somenewpropositionsthatweredifferentfromEuclideangeometrywerederived,whichledtotheemergenceofnon-Euclideangeometry.Euclidreliedonlyonformalthinkingandformedasystemthroughgraphicalevidence,anduseditfirstinthe29thproposition.Duetocertaincontradictionsofthefifthpostulateanditsindependencefromotheraxioms,definitions,andpostulates,mathematiciansofthepastdynastiesactivelyexplored,triedtomakeupforit,ortriedtoexpandnewfields,whichgreatlydevelopedmathematics.Themostsignificanceofthefifthpostulatetothedevelopmentofgeometrycomesfromthenegationofthefifthpostulate,thatis,thebirthofnon-euclideangeometry.KeyWords：Mathematics,Thefifthpostulate,Non-euclideangeometry,GeometryIII目录1第五公设的内容及背景..............................................11.1主要内容:.......................................................11.2产生背景：......................................................22第五公设的历史证明................................................32.1普罗克鲁斯的试证................................................42.2萨凯里的试证....................................................52.3兰伯特的试证....................................................62.3其他数学家的试证................................................73欧几里得第五公设对数学的影响......................................83.1非欧几何的创立..................................................94非欧几何的分支及发展.............................................104.1罗氏几何学.....................................................114.2黎曼几何.......................................................114.3三种几何的关系.................................................13参考文献：.........................................................14致谢...........................................................1511第五公设的内容及背景1.1主要内容:若是两条直线都能与第三条直线相交，并且在同一边的内角之和小于两个直角，那么这两条直线在这一边必定相交[1]。1.2产生背景：几何学是最先能被应用在土地测量，这种计算和测量土地的方式被引到希腊时，希腊人就把它抽象成了一种学科.希腊哲学家亚里士多德（aristole...
