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1、ntheHypotheseswhihlieattheBasesofGeometry.BernhardRiemannTranslatedbyWilliamingdonCliord[ature,Vol.V.os.183,184,pp.14{17,36,37.℄TransribedbyD.R.WilkinsreliminaryVersion:Deember1998ntheHypotheseswhihlieattheBasesofGeometry.BernhardRiemannTranslatedbyWilliam
2、ingdonCliord[ature,Vol.V.os.183,184,pp.14{17,36,37.℄lanofthenvestigation.tisknownthatgeometryassumes,asthingsgiven,boththenotionofspaeandtherstpriniplesofonstrutionsinspae.Shegivesdenitionsofthemwhiharemerelynominal,whilethetruedeterminationsappearintheformo
3、faxioms.Therelationoftheseassumptionsremainsonsequentlyindarkness;weneitherpereivewhetherandhowfartheironnetionisneessary,norapriori,whetheritispossible.FromEulidtoegendre(tonamethemostfamousofmodernreform-inggeometers)thisdarknesswaslearedupneitherbymathematiiansn
4、orbysuhphilosophersasonernedthemselveswithit.Thereasonofthisisdoubtlessthatthegeneralnotionofmultiplyextendedmagnitudes(inwhihspae-magnitudesareinluded)remainedentirelyunworked.haveintherstplae,therefore,setmyselfthetaskofonstrutingthenotionofamultiplyextendedmagni
5、tudeoutofgeneralnotionsofmagnitude.twillfollowfromthisthatamultiplyextendedmagnitudeisapableofdierentmeasure-relations,andonsequentlythatspaeisonlyapartiularaseofatriplyextendedmagnitude.Butheneowsasaneessaryonsequenethatthepropositionsofgeometryannotbederivedfromg
6、eneralnotionsofmagnitude,butthatthepropertieswhihdistinguishspaefromotheron-eivabletriplyextendedmagnitudesareonlytobededuedfromexperiene.Thusarisestheproblem,todisoverthesimplestmattersoffatfromwhihthemeasure-relationsofspaemaybedetermined;aproblemwhihfromthenature
7、oftheaseisnotompletelydeterminate,sinetheremaybeseveralsystemsofmattersoffatwhihsuÆetodeterminethemeasure-relationsofspaethemostimportantsystemforourpresentpurposebeingthatwhihEulidhaslaiddownasafoundation.Thesemattersoffatarelikeall1mattersoffatnotneessary,butonlyo
8、fempirialertainty;theyarehy-potheses.Wemaythereforeinvestigatetheirprobability,whihwithinthelimitsofobservationisofourseverygreat,andinqui
