圆周率计算的四个时期

[]2000-07-05[]张晓贵(1965-),男,安徽肥东人,台州师范专科学校数学系讲师张晓贵(,317007)[]本文展示了圆周率计算的四个时期:(1)经验性获得时期(2)几何推算时期(3)解析计算时期(4)计算机运算时期[]圆周率;P;计算;时期[]O12[]A[]1003-191X(2000)05-0066-04,,,,P,P,,,P,/9(9),?,1,8,88,64,0,,,256/81,3.1605,,,,P3,25/83.125P2000,,/0,,Hippocrates,Anaxagoras,Antiphon,Eudoxus,,56,,416,ViruviusPollio,,P=3.125,22/7,,56,/0,(220)(/0)第17卷第5期2000年9月辽宁教育学院学报JournalofLiaoningEducationalInstituteVol.17No.5Sep2000P=10,56,,,,,,56,,P:P=4(1-215)2=3.0044...P=4[1+13(2-1)]2=3.0883...,3,PArchimedes(287-212BC),223/7122/7,96Archimedes1,3@2n-1,anbn,n=2na1,a2,,,an,,,b1,b2,,bn,,P,,an=Ktan(P/K),bn=Ksin(P/K)an+1=2Ktan(P/2K),bn=2Ksin(P/2K)K=3@2n-1,(1)...(1/an+1/bn)=2/an+1(2)...an+1bn=(bn+1)2a1=3tan(P3)=33,b1=3sin(P3)=332(1)a2,(2)b2,(1)a3,(2)b3,a6b6a6Pb6,Archimedes,(),a1,b1a6,b6,150,Ptolemy,1/60,60,5011250,360(360@1250)A120,P=3+860+30602=377120=3.1416Archimedes500)1000,,,3.1416,56,:1004,8,62000,20000,3.1416,,,,67张晓贵:圆周率计算的四个时期1119,/39271250,,227,0,:3.1416=39271250P227U3.1429,,n2n,Lnn,Snn,S,L,r:Ln@rA2=S2nyS,LnyL(ny])S=L@rA2192314.1024100P314.2704157/50=3.14,/0S2nSSn+2(S2n-Sn),,,,:3.1415926P3.141592756,:/,,,0,al-KashiViete,,,1600VanCeulen35,P,Willis:2P=1#3#3#5#5#7,2#2#4#4#6#6,:P4=1-13+15-17+,Leibniz(1646-1716),JamesGregory(1638-1675),,P,P,,Gregory,P4,0.00005=1/20000,10000GregoryP,tan-1x=x-x33+x55-...(-1[x[1)x=1,tan-1(13)=P6Gregorg,P6=13(1-13#3+15#3#3-17#3#3#3+...),10119.39#30.00005,9,4,P,1/21/3Gregory68辽宁教育学院学报2000年第5期P4=tan-1(12)+tan-1(13)P4=tan-1(1a)+tan-1(1b)ab,,1706Machin:P4=4tan-1(15)-tan-1[1239]P1647,Oughtredd/P,1697,DavidGregoryP/r,PWilliamJones1706,Euler1737,,,P,PShanks,707,1837,1945,Ferguson572Shanks,:P=4-4(12#13+12#4#15+1#32#4#6#17+...)ShanksP1761Lambert,ShanksP,LindemannP,P,/0,,,,Shanks527,,,1949ENIAC70P2037,1999Kana-daTakahashi73P68,719,470,000,,,,,Kanada:/P,0[][1]JJO.ConnorandEFRobertson.Pithroughtheages.~history/HistTopics/Pi_through_the_ages.html.[2]朱志学等1数学的历史思想和方法[M]1哈尔滨:哈尔滨出版社,1990.[3]斯科特1数学史[M]1北京:商务印书馆,1981.[4]梁宗巨1数学历史典故[M]1沈阳:辽宁教育出版社,1995.69张晓贵:圆周率计算的四个时期