数学专业英语2.docx

数学专业英语2.docx

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数学专业英语2

MathematicalEnglish

Dr.XiaominZhang

Email:

zhangxiaomin@

§2.2GeometryandTrigonometry

TEXTAWhystudygeometry?

Whydowestudygeometry?

Thestudentbeginningthestudyofthistextmaywellask,“Whatisgeometry?

WhatcanIexpecttogainfromthisstudy?

”

Manyleadinginstitutionsofhigherlearninghaverecognizedthatpositivebenefitscanbegainedbyallwhostudythisbranchofmathematics.Thisisevidentfromthefactthattheyrequirestudyofgeometryasaprerequisitetomatriculationinthoseschools.

GeometryhaditsoriginlongagointhemeasurementsbytheBabyloniansandEgyptiansoftheirlandsinundatedbythefloodsoftheNileRiver.TheGreekwordgeometryisderivedfromgeo,meaning“earth”,andmetron,meaning“measure”.Asearlyas2000B.C.wefindthelandsurveyorsofthesepeoplere-establishingvanishinglandmarksandboundariesbyutilizingthetruthsofgeometry.

Geometryisasciencethatdealswithformsmadebylines.Astudyofgeometryisanessentialpartofthetrainingofthesuccessfulengineer,scientist,architect,anddraftsman.Thecarpenter,machinist,stonecutter,artist,anddesignerallapplythefactsofgeometryintheirtrades.Inthiscoursethestudentwilllearnagreatdealaboutgeometricfiguressuchaslines,angles,triangles,circles,anddesignsandpatternsofmanykinds.

Oneofthemostimportantobjectivesderivedfromastudyofgeometryismakingthestudentbemorecriticalinhislistening,reading,andthinking.Instudyinggeometryheisledawayfromthepracticeofblindacceptanceofstatementsandideasandistaughttothinkclearlyandcriticallybeforeformingconclusions.

Therearemanyotherlessdirectbenefitsthestudentofgeometrymaygain.AmongtheseonemustincludetrainingintheexactuseoftheEnglishlanguageandintheabilitytoanalyzeanewsituationorproblemintoitsbasicparts,andutilizingperseverance,originality,andlogicalreasoninginsolvingtheproblem.Anappreciationfortheorderlinessandbeautyofgeometricformsthataboundinman’sworksandofthecreationsofnaturewillbeabyproductofthestudyofgeometry.Thestudentshouldalsodevelopanawarenessofthecontributionsofmathematicsandmathematicianstoourcultureandcivilization.

TEXTBSomegeometricalterms

1.Solidsandplanes.Asolidisathree-dimensionalfigure.Commonexamplesofsolidarecube,sphere,cylinder,coneandpyramid.

Acubehassixfaceswhicharesmoothandflat.Thesefacesarecalledplanesurfacesorsimplyplanes.Aplanesurfacehastwodimensions,lengthandwidth.Thesurfaceofablackboardoratabletopisanexampleofaplanesurface.

2.Linesandlinesegments.Weareallfamiliarwithlines,butitisdifficulttodefinetheterm.Alinemayberepresentedbythemarkmadebymovingapencilorpenacrossapieceofpaper.Alinemaybeconsideredashavingonlyonedimension,length.Althoughwhenwedrawalinewegiveitbreadthandthickness,wethinkonlyofthelengthofthetracewhenconsideringtheline.Apointhasnolength,nowidth,andnothickness,butmarksaposition.Wearefamiliarwithsuchexpressionsaspencilpointandneedlepoint.Werepresentapointbyasmalldotandnameitbyacapitalletterprintedbesideit,as“pointA”inFig.2-2-1.

Thelineisnamedbylabelingtwopointsonitwithcapitallettersoronesmallletternearit.ThestraightlineinFig.2-2-2isread“lineAB”or“linel”.Astraightlineextendsinfinitelyfarintwodirectionsandhasnoends.Thepartofthelinebetweentwopointsonthelineistermedalinesegment.Alinesegmentisnamedbythetwoendpoints.Thus,inFig.2-2-2,werefertoABaslinesegmentoflinel.Whennoconfusionmayresult,theexpression“linesegmentAB”isoftenreplacedbysegmentABor,simply,lineAB.

Therearethreekindsoflines:

thestraightline,thebrokenline,andthecurvedline.Acurvedlineor,simply,curveislinenopartofwhichisstraight.Abrokenlineiscomposedofjoined,straightlinesegments,asABCDEofFig.2-2-3.

3.Partsofacircle.Acircleisaclosedcurvelyinginoneplane,allpointsofwhichareequidistantfromafixedpointcalledthecenter（Fig.2-2-4）.Thesymbolforacircleis⊙.InFig.2-2-4,Oisthecenterof⊙ABC,orsimplyof⊙O.Alinesegmentdrawnfromthecenterofthecircletoapointonthecircleisaradius（plural,radii）ofthecircle.OA,OB,andOCareradiiof⊙O,Adiameterofacircleisalinesegmentthroughthecenterofthecirclewithendpointsonthecircle.Adiameterisequaltotworadii.Achordisanylinesegmentjoiningtwopointsonthecircle.EDisachordofthecircleinFig.2-2-4.

Fromthisdefinitionisshouldbeapparentthatadiameterisachord.Anypartofacircleisanarc,suchasarcAE.PointsAandEdividethecircleintominorarcAEandmajorarcABE.Adiameterdividesacircleintotwoarcstermedsemicircles,suchasAB.Thecircumferenceisthelengthofacircle.

SUPPLEMENTARuler-and-compassconstructions

Anumberofancientproblemsingeometryinvolvetheconstructionoflengthsoranglesusingonlyanidealisedrulerandcompass.Therulerisindeedastraightedge,andmaynotbemarked;thecompassmayonlybesettoalreadyconstructeddistances,andusedtodescribecirculararcs.

Somefamousruler-and-compassproblemshavebeenprovedimpossible,inseveralcasesbytheresultsofGaloistheory.Inspiteoftheseimpossibilityproofs,somemathematicalamateurspersistintryingtosolvetheseproblems.Manyofthemfailtounderstandthatmanyoftheseproblemsaretriviallysolubleprovidedthatothergeometrictransformationsareallowed:

forexample,squaringthecircleispossibleusinggeometricconstructions,butnotpossibleusingrulerandcompassesalone.MathematicianUnderwoodDudleyhasmadeasidelineofcollectingfalseruler-and-compassproofs,aswellasotherworkbymathematicalcranks,andhascollectedthemintoseveralbooks.

SquaringthecircleThemostfamousoftheseproblems,“squaringthecircle”,involvesconstructingasquarewiththesameareaasagivencircleusingonlyrulerandcompasses.Squaringthecirclehasbeenprovedimpossible,asitinvolvesgeneratingatranscendentalratio,namely1:

√π.

Onlyalgebraicratioscanbeconstructedwithrulerandcompassesalone.Thephrase“squaringthecircle”isoftenusedtomean“doingtheimpossible”forthisreason.Withouttheconstraintofrequiringsolutionbyrulerandcompassesalone,theproblemiseasilysolublebyawidevarietyofgeometricandalgebraicmeans,andhasbeensolvedmanytimesinantiquity.

DoublingthecubeUsingonlyrulerandcompasses,constructthesideofacubethathastwicethevolumeofacubewithagivenside.Thisisimpossiblebecausethecuberootof2,thoughalgebraic,cannotbecomputedfromintegersbyaddition,subtraction,multiplication,division,andtakingsquareroots.

AngletrisectionUsingonlyrulerandcompasses,constructananglethatisone-thirdofagivenarbitraryangle.Thisrequirestakingthecuberootofanarbitrarycomplexnumberwithabsolutevalue1andislikewiseimpossible.

Constructingwithonlyruleroronlycompass

Itispossible,asshownbyGeorgMohr,toconstructanythingwithjustacompassthatcanbeconstructedwithrulerandcompass.Itisimpossibletotakeasquarerootwithjustaruler,sosomethingscannotbeconstructedwitharulerthatcanbeconstructedwithacompass;butgivenacircleanditscenter,theycanbeconstructed.

ProblemHowcanyoudeterminethemidpointofanygivenlinesegmentwithonlycompass?

SUPPLEMENTBArchimedesandOntheSphereandtheCylinder

Archimedes（287BC-212BC）wasanAncientmathematician,astronomer,philosopher,physicistandengineerbornintheGreekseaportcolonyofSyracuse.Heisconsideredbysomemathhistorianstobeoneofhistory'sgreatestmathematicians,alongwithpossiblyNewton,GaussandEuler.

Hewasanaristocrat,thesonofanastronomer,butlittleisknownofhisearlylifeexceptthathestudiedunderfollowersofEuclidinAlexandria,EgyptbeforereturningtohisnativeSyracuse,thenanindependentGreekcity-state.SeveralofhisbookswerepreservedbytheGreeksandArabsintotheMiddleAges,and,fortunately,theRomanhistorianPlutarchdescribedafewepisodesfromhislife.Inmanyareasofmathematicsaswellasinhydrostaticsandstatics,hisworkandresultswerenotsurpassedforover1500years!

Heapproximatedtheareaofcircles（andthevalueof¼）bysummingtheareasofinscribedandcircumscribedrectangles,andgeneralizedthis"methodofexhaustion,"bytakingsmallerandsmallerrectangularareasandsummingthem,tofindtheareasandevenvolumesofseveralothershapes.ThisanticipatedtheresultsofthecalculusofNewtonandLeibnizbyalmost2000years!

Hefoundtheareaandtangentstothecurvetracedbyapointmovingwithuniformspeedalongastraightlinewhichisrevolvingwithuniformangularspeedaboutafixedpoint.Thiscurve,describedbyr=a

inpolarcoordinates,isnowcalledthe"spiralofArchimedes."Withcalculusitisaneasyproblem;withoutcalculusitisverydifficult.

ThekingofSyracuseonceaskedArchimedestofindawayofdeterminingifoneofhiscrownswaspuregoldwithoutdestroyingthecrownintheprocess.Thecrownweighedthecorrectamountbutthatwasnotaguaranteethatitwaspuregold.ThestoryistoldthatasArchimedesloweredhimselfintoabathhenoticedthatsomeofthewaterwasdisplacedbyhisbodyandflowedovertheedgeofthetub.Thiswasjusttheinsightheneededtorealizethatthecrownshouldnotonlyweightherightamountbutshoulddisplacethesamevolumeasanequalw