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1、BOOKREVIEWSBULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume37,Number1,Pages77{78S0273-0979(99)00806-XArticleelectronicallypublishedonDecember21,1999Poincare'sreviewofHilbert'sFoundationsofgeometry,byDavidHilbertWhatarethefundamentalprinciplesofgeometry?whatisitsorigin?itsnature?itssc
2、ope?Thesearequestionswhichhaveatalltimesengagedtheattentionofmathematiciansandthinkers,butwhichaboutacenturyagotookonanentirelynewaspect,thankstotheideasofLobachevskyandofBolyai.Foralongtimeweattemptedtodemonstratethepropositionknownasthepos-tulateofEuclid;weconstantlyfailed;weknownowthereason
3、forthesefailures.Lobachevskysucceededinbuildingalogicaledi ceascoherentasthegeometryofEuclid,butinwhichthefamouspostulateisassumedfalse,andinwhichthesumoftheanglesofatriangleisalwayslessthantworightangles.Riemanndevisedanotherlogicalsystem,equallyfreefromcontradiction,inwhichthissumisontheothe
4、rhandalwaysgreaterthantworightangles.Thesetwogeometries,thatofLobachevskyandthatofRiemann,arewhatarecalledthenon-euclideangeome-tries.ThepostulateofEuclidthencannotbedemonstrated;andthisimpossibilityisasabsolutelycertainasanymathematicaltruthwhatsoever
5、afactwhichdoesnotpreventtheAcademiedesSc
6、iencesfromreceivingeveryyearseveralnewproofs,towhichitnaturallyrefusesthehospitalityoftheComptesrendus.Muchhasalreadybeenwrittenonthenon-euclideangeometries;oncetheyscan-dalizedus;nowwehavebecomeaccustomedtotheirparadoxes;somepeoplehavegonesofarastodoubtthetruthofthepostulateandtoaskwhetherrea
7、lspaceisplane,asEuclidassumed,orwhetheritmaynotpresentaslightcurvature.Theyevensupposedthatexperimentcouldgivethemananswer[250]tothisquestion.Needlesstoaddthatthiswasatotalmisconceptionofthenatureofgeometry,whichisnotanexperimentalscience.Butwhy,amongalltheaxiomsofgeometry,shouldthispostulateb
8、etheonlyonewhichcouldbedeniedwithouto encetologic?Whenceshoulditderivethisprivilege?Thereseemstobenogoodreasonforthis,andmanyotherconceptionsarepossible.However,manycontemporarygeometersdonotappeartothinkso.Inrecogniz-ingtheclaimsofthet
