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1、OntheHypotheseswhichlieattheBasesofGeometry.BernhardRiemannTranslatedbyWilliamKingdonCli®ord[Nature,Vol.VIII.Nos.183,184,pp.14{17,36,37.]TranscribedbyD.R.WilkinsPreliminaryVersion:December1998OntheHypotheseswhichlieattheBasesofGeometry.BernhardRiemannTranslat
2、edbyWilliamKingdonCli®ord[Nature,Vol.VIII.Nos.183,184,pp.14{17,36,37.]PlanoftheInvestigation.Itisknownthatgeometryassumes,asthingsgiven,boththenotionofspaceandthe¯rstprinciplesofconstructionsinspace.Shegivesde¯nitionsofthemwhicharemerelynominal,whilethetruede
3、terminationsappearintheformofaxioms.Therelationoftheseassumptionsremainsconsequentlyindarkness;weneitherperceivewhetherandhowfartheirconnectionisnecessary,norapriori,whetheritispossible.FromEuclidtoLegendre(tonamethemostfamousofmodernreform-inggeometers)thisd
4、arknesswasclearedupneitherbymathematiciansnorbysuchphilosophersasconcernedthemselveswithit.Thereasonofthisisdoubtlessthatthegeneralnotionofmultiplyextendedmagnitudes(inwhichspace-magnitudesareincluded)remainedentirelyunworked.Ihaveinthe¯rstplace,therefore,set
5、myselfthetaskofconstructingthenotionofamultiplyextendedmagnitudeoutofgeneralnotionsofmagnitude.Itwillfollowfromthisthatamultiplyextendedmagnitudeiscapableofdi®erentmeasure-relations,andconsequentlythatspaceisonlyaparticularcaseofatriplyextendedmagnitude.Buthe
6、nce°owsasanecessaryconsequencethatthepropositionsofgeometrycannotbederivedfromgeneralnotionsofmagnitude,butthatthepropertieswhichdistinguishspacefromothercon-ceivabletriplyextendedmagnitudesareonlytobededucedfromexperience.Thusarisestheproblem,todiscoverthesi
7、mplestmattersoffactfromwhichthemeasure-relationsofspacemaybedetermined;aproblemwhichfromthenatureofthecaseisnotcompletelydeterminate,sincetheremaybeseveralsystemsofmattersoffactwhichsu±cetodeterminethemeasure-relationsofspace
8、themostimportantsystemforourprese
9、ntpurposebeingthatwhichEuclidhaslaiddownasafoundation.Thesemattersoffactare
10、likeall1mattersoffact
11、notnecessary,butonlyofempiricalcertainty;theyarehy-potheses.Wemaythereforeinvestigatethei
