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1、BOOKREVIEWSBULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume37,Number1,Pages77{78S0273-0979(99)00806-XArticleelectronicallypublishedonDecember21,1999Poincare'sreviewofHilbert'sFoundationsofgeometry,byDavidHilbertWhatarethefundamentalprinciplesofgeometr
2、y?whatisitsorigin?itsnature?itsscope?Thesearequestionswhichhaveatalltimesengagedtheattentionofmathematiciansandthinkers,butwhichaboutacenturyagotookonanentirelynewaspect,thankstotheideasofLobachevskyandofBolyai.Foralongtimeweattemptedtodemonstratetheproposition
3、knownasthepos-tulateofEuclid;weconstantlyfailed;weknownowthereasonforthesefailures.Lobachevskysucceededinbuildingalogicaledi ceascoherentasthegeometryofEuclid,butinwhichthefamouspostulateisassumedfalse,andinwhichthesumoftheanglesofatriangleisalwayslessthantwori
4、ghtangles.Riemanndevisedanotherlogicalsystem,equallyfreefromcontradiction,inwhichthissumisontheotherhandalwaysgreaterthantworightangles.Thesetwogeometries,thatofLobachevskyandthatofRiemann,arewhatarecalledthenon-euclideangeome-tries.ThepostulateofEuclidthencann
5、otbedemonstrated;andthisimpossibilityisasabsolutelycertainasanymathematicaltruthwhatsoever
6、afactwhichdoesnotpreventtheAcademiedesSciencesfromreceivingeveryyearseveralnewproofs,towhichitnaturallyrefusesthehospitalityoftheComptesrendus.Muchhasalreadybeenwritteno
7、nthenon-euclideangeometries;oncetheyscan-dalizedus;nowwehavebecomeaccustomedtotheirparadoxes;somepeoplehavegonesofarastodoubtthetruthofthepostulateandtoaskwhetherrealspaceisplane,asEuclidassumed,orwhetheritmaynotpresentaslightcurvature.Theyevensupposedthatexper
8、imentcouldgivethemananswer[250]tothisquestion.Needlesstoaddthatthiswasatotalmisconceptionofthenatureofgeometry,whichisnotanexperimentalscience.Butwhy,amongalltheaxiomsofgeometry,shouldthispostulatebetheonlyonewhichcouldbedeniedwithouto encetologic?Whenceshouldi
9、tderivethisprivilege?Thereseemstobenogoodreasonforthis,andmanyotherconceptionsarepossible.However,manycontemporarygeometersdonotappeartothinkso.Inrecogniz-ingtheclaimsofthet
