高斯JohannCarlFriedrichGauss.docx

高斯JohannCarlFriedrichGauss.docx

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高斯JohannCarlFriedrichGauss

JohannCarlFriedrichGauss

Born:

30April1777inBrunswick,DuchyofBrunswick（nowGermany）

Died:

23Feb1855inGöttingen,Hanover（nowGermany）

Attheageofseven,CarlFriedrichGaussstartedelementaryschool,andhispotentialwasnoticedalmostimmediately.Histeacher,Büttner,andhisassistant,MartinBartels,wereamazedwhenGausssummedtheintegersfrom1to100instantlybyspottingthatthesumwas50pairsofnumberseachpairsummingto101.

In1788GaussbeganhiseducationattheGymnasiumwiththehelpofBüttnerandBartels,wherehelearntHighGermanandLatin.AfterreceivingastipendfromtheDukeofBrunswick-Wolfenbüttel,GaussenteredBrunswickCollegiumCarolinumin1792.AttheacademyGaussindependentlydiscoveredBode'slaw,thebinomialtheoremandthearithmetic-geometricmean,aswellasthelawofquadraticreciprocityandtheprimenumbertheorem.

In1795GaussleftBrunswicktostudyatGöttingenUniversity.Gauss'steachertherewasKästner,whomGaussoftenridiculed.HisonlyknownfriendamongstthestudentswasFarkasBolyai.Theymetin1799andcorrespondedwitheachotherformanyyears.

GaussleftGöttingenin1798withoutadiploma,butbythistimehehadmadeoneofhismostimportantdiscoveries-theconstructionofaregular17-gonbyrulerandcompassesThiswasthemostmajoradvanceinthisfieldsincethetimeofGreekmathematicsandwaspublishedasSectionVIIofGauss'sfamouswork,DisquisitionesArithmeticae.

GaussreturnedtoBrunswickwherehereceivedadegreein1799.AftertheDukeofBrunswickhadagreedtocontinueGauss'sstipend,herequestedthatGausssubmitadoctoraldissertationtotheUniversityofHelmstedt.HealreadyknewPfaff,whowaschosentobehisadvisor.Gauss'sdissertationwasadiscussionofthefundamentaltheoremofalgebra.

Withhisstipendtosupporthim,Gaussdidnotneedtofindajobsodevotedhimselftoresearch.HepublishedthebookDisquisitionesArithmeticaeinthesummerof1801.Thereweresevensections,allbutthelastsection,referredtoabove,beingdevotedtonumbertheory.

InJune1801,Zach,anastronomerwhomGausshadcometoknowtwoorthreeyearspreviously,publishedtheorbitalpositionsofCeres,anew"smallplanet"whichwasdiscoveredbyGPiazzi,anItalianastronomeron1January,1801.Unfortunately,Piazzihadonlybeenabletoobserve9degreesofitsorbitbeforeitdisappearedbehindtheSun.Zachpublishedseveralpredictionsofitsposition,includingonebyGausswhichdifferedgreatlyfromtheothers.WhenCereswasrediscoveredbyZachon7December1801itwasalmostexactlywhereGausshadpredicted.Althoughhedidnotdisclosehismethodsatthetime,Gausshadusedhisleastsquaresapproximationmethod.

InJune1802GaussvisitedOlberswhohaddiscoveredPallasinMarchofthatyearandGaussinvestigateditsorbit.OlbersrequestedthatGaussbemadedirectoroftheproposednewobservatoryinGöttingen,butnoactionwastaken.GaussbegancorrespondingwithBessel,whomhedidnotmeetuntil1825,andwithSophieGermain.

GaussmarriedJohannaOstoffon9October,1805.Despitehavingahappypersonallifeforthefirsttime,hisbenefactor,theDukeofBrunswick,waskilledfightingforthePrussianarmy.In1807GaussleftBrunswicktotakeupthepositionofdirectoroftheGöttingenobservatory.

GaussarrivedinGöttingeninlate1807.In1808hisfatherdied,andayearlaterGauss'swifeJohannadiedaftergivingbirthtotheirsecondson,whowastodiesoonafterher.GausswasshatteredandwrotetoOlbersaskinghimtogivehimahomeforafewweeks,

togathernewstrengthinthearmsofyourfriendship-strengthforalifewhichisonlyvaluablebecauseitbelongstomythreesmallchildren.

Gausswasmarriedforasecondtimethenextyear,toMinnathebestfriendofJohanna,andalthoughtheyhadthreechildren,thismarriageseemedtobeoneofconvenienceforGauss.

Gauss'sworkneverseemedtosufferfromhispersonaltragedy.Hepublishedhissecondbook,TheoriamotuscorporumcoelestiuminsectionibusconicisSolemambientium,in1809,amajortwovolumetreatiseonthemotionofcelestialbodies.Inthefirstvolumehediscusseddifferentialequations,conicsectionsandellipticorbits,whileinthesecondvolume,themainpartofthework,heshowedhowtoestimateandthentorefinetheestimationofaplanet'sorbit.Gauss'scontributionstotheoreticalastronomystoppedafter1817,althoughhewentonmakingobservationsuntiltheageof70.

MuchofGauss'stimewasspentonanewobservatory,completedin1816,buthestillfoundthetimetoworkonothersubjects.HispublicationsduringthistimeincludeDisquisitionesgeneralescircaserieminfinitam,arigoroustreatmentofseriesandanintroductionofthehypergeometricfunction,Methodusnovaintegraliumvaloresperapproximationeminveniendi,apracticalessayonapproximateintegration,BestimmungderGenauigkeitderBeobachtungen,adiscussionofstatisticalestimators,andTheoriaattractioniscorporumsphaeroidicorumellipticorumhomogeneorummethodusnovatractata.Thelatterworkwasinspiredbygeodesicproblemsandwasprincipallyconcernedwithpotentialtheory.Infact,Gaussfoundhimselfmoreandmoreinterestedingeodesyinthe1820s.

Gausshadbeenaskedin1818tocarryoutageodesicsurveyofthestateofHanovertolinkupwiththeexistingDanishgrid.Gausswaspleasedtoacceptandtookpersonalchargeofthesurvey,makingmeasurementsduringthedayandreducingthematnight,usinghisextraordinarymentalcapacityforcalculations.HeregularlywrotetoSchumacher,OlbersandBessel,reportingonhisprogressanddiscussingproblems.

Becauseofthesurvey,GaussinventedtheheliotropewhichworkedbyreflectingtheSun'sraysusingadesignofmirrorsandasmalltelescope.However,inaccuratebaselineswereusedforthesurveyandanunsatisfactorynetworkoftriangles.Gaussoftenwonderedifhewouldhavebeenbetteradvisedtohavepursuedsomeotheroccupationbuthepublishedover70papersbetween1820and1830.

In1822GausswontheCopenhagenUniversityPrizewithTheoriaattractionis...togetherwiththeideaofmappingonesurfaceontoanothersothatthetwoaresimilarintheirsmallestparts.Thispaperwaspublishedin1825andledtothemuchlaterpublicationofUntersuchungenüberGegenständederHöherenGeodäsie（1843and1846）.ThepaperTheoriacombinationisobservationumerroribusminimisobnoxiae（1823）,withitssupplement（1828）,wasdevotedtomathematicalstatistics,inparticulartotheleastsquaresmethod.

Fromtheearly1800sGausshadaninterestinthequestionofthepossibleexistenceofanon-Euclideangeometry.HediscussedthistopicatlengthwithFarkasBolyaiandinhiscorrespondencewithGerlingandSchumacher.Inabookreviewin1816hediscussedproofswhichdeducedtheaxiomofparallelsfromtheotherEuclideanaxioms,suggestingthathebelievedintheexistenceofnon-Euclideangeometry,althoughhewasrathervague.GaussconfidedinSchumacher,tellinghimthathebelievedhisreputationwouldsufferifheadmittedinpublicthathebelievedintheexistenceofsuchageometry.

In1831FarkasBolyaisenttoGausshissonJánosBolyai'sworkonthesubject.Gaussreplied

topraiseitwouldmeantopraisemyself.

Again,adecadelater,whenhewasinformedofLobachevsky'sworkonthesubject,hepraisedits"genuinelygeometric"character,whileinalettertoSchumacherin1846,statesthathe

hadthesameconvictionsfor54years

indicatingthathehadknownoftheexistenceofanon-Euclideangeometrysincehewas15yearsofage（thisseemsunlikely）.

Gausshadamajorinterestindifferentialgeometry,andpublishedmanypapersonthesubject.Disquisitionesgeneralescircasuperficiescurva（1828）washismostrenownedworkinthisfield.Infact,thispaperrosefromhisgeodesicinterests,butitcontainedsuchgeometricalideasasGaussiancurvature.ThepaperalsoincludesGauss'sfamoustheoremaegregrium:

IfanareainE3canbedeveloped（i.e.mappedisometrically）intoanotherareaofE3,thevaluesoftheGaussiancurvaturesareidenticalincorrespondingpoints.

Theperiod1817-1832wasaparticularlydistressingtimeforGauss.Hetookinhissickmotherin1817,whostayeduntilherdeathin1839,whilehewasarguingwithhiswifeandherfamilyaboutwhethertheyshouldgotoBerlin.HehadbeenofferedapositionatBerlinUniversityandMinnaandherfamilywerekeentomovethere.Gauss,however,neverlikedchangeanddecidedtostayinGöttingen.In1831Gauss'ssecondwifediedafteralongillness.

In1831,WilhelmWeberarrivedinGöttingenasphysicsprofessorfillingTobiasMayer'schair.GausshadknownWebersince1828andsupportedhisappointment.Gausshadworkedonphysicsbefore1831,publishingÜbereinneuesallgemeinesGrundgesetzderMechanik,whichcontainedtheprincipleofleastconstraint,andPrincipiageneraliatheoriaefiguraefluidoruminstatuaequilibriiwhichdiscussedforcesofattraction.ThesepaperswerebasedonGauss'spotentialtheory,whichprovedofgreatimportanceinhisworkonphysics.Helatercametobelievehispotentialtheoryandhismethodofleastsquaresprovidedvitallinksbetweenscienceandnature.

In1832,GaussandWeberbeganinvestigatingthetheoryofterrestrialmagnetismafterAlexandervonHumboldtattemptedtoobtainGauss'sassistanceinmakingagridofmagneticobservationpointsaroundtheEarth.Gausswasexcitedbythisprospectandby1840hehadwrittenthreeimportantpapersonthesubject:

Intensitasvismagneticaeterrestrisadmensuramabsolutamrevocata（1832）,AllgemeineTheoriedesErdmagnetismus（1839）and