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从数学史角度谈三棱锥体积公式的证明程汉波杨春波(,430079),V=13Sh,:56;;;.,,.1Ac-ABC,vABCS,h,V=13Sh.1.1,(56Ü,34),,.图11,Ac-ABC,,,VDEF-GIC=SvGIC@hF=14S@12h=18Sh;VDGH-EIB=12SGHBIhD=12@12S@12h=18Sh,V1=18Sh+18Sh=14Sh.,V2=2@14SvAHGhD=2@14@14S@12h=142Sh.,,,,,V=(14+142+143+,+14n+,)Sh=14/(1-14)Sh=13Sh..,,,.1.21800,/,0,/0)))/,013.,..,S,h,V1,V2.A,,A,B,AB.A59#课外园地#数学通讯)2012年第6期(下半月),S1,S2,h1,S1S=S2S=(h1h)2,S1=S2,V1=V2...2,vABC,AAcABC-AcBcCc,AcC,BcC,,V1,V2,V3.C-AcABC-AcBBc,SvAcAB=SvAcBBc,C,,V1=V2.V2=V3,V1=V2=V3,V=V1=13Sh..图2,/0.1.3,,/0,/0,.,?,:Ac-ABCn,n-1,kSk,Vk,SkS=(h-knhh)2,Sk=(n-kn)2S,Vk=Skhn=(n-k)2n3Sh.V(n)=6n-1k=1Vk=6n-1k=1(n-k)2n3Sh=n(n-1)(2n-1)6n3Sh,V=limny]V(n)=limny]n(n-1)(2n-1)6n3#Sh=13Sh../,,0,,.1.4,,,.,x,x,S(x),xI[0,h].S(x)S=(xh)2,S(x)=x2h2S,V=Qh0S(x)dx=Qh0x2h2Sdx=13Sh..,.2,,.2.1nS,hn(n\3),n-2,n-2,S1,S2,S3,,,Sn-2,nV=6n-2k=1Vk=6n-2k=113#Skh=13h6n-2k=1Sk=13Sh.S,h,n,n,:V=13Sh.2.2R,nA1A2,An(n\3)()NA1,NA2,,,60数学通讯)2012年第6期(下半月)#课外园地#NAn,Sn,OA1,A2,,,AnnO-A1A2,An.n,,nO-A1A2,AnV=613SiR=13R6Si=13SnR=13R3[NA1+NA2+,+NAn-(n-2)P].3:/,0.,,.,,,.:[1]().[M].:,2009.[2]JohnStillwell(,).[M].:,2011.[3].()[M].:,2009.[4],.[M].:,2011.(:2012-03-22)测字与数学*陈荣杨飞(,400030)*/#0:/0(2011-KG-046),,.,,,.,,.,5656,56,.,,.1,,A1+A2+A3+,+An=M.,,./0,,:,,.,-.,.,,,:/,,,61#课外园地#数学通讯)2012年第6期(下半月)