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1、THEELEMENTSOFNON-EUCLIDEANGEOMETRYBYJULIANLOWELLCOOLIDGEPH.D.ASSISTANTPROFESSOROFMATHEMATICSINHARVARDUNIVERSITYOXFORDATTHECLARENDONPRESS1909HENRYFROWDE,M.A.PUBLISHERTOTHEUNIVERSITYOFOXFORDLONDON,EDINBURGH,NEWYORKTORONTOANDMELBOURNEPREFACETHEheroicageofnon-
2、euclideangeometryispassed.ItislongsincethedayswhenLobatchewskytimidlyreferredtohissystemasan*imaginarygeometry',andthenewsubjectappearedasadangerouslapsefromtheorthodoxdoctrineofEuclid.Theattempttoprovetheparallelaxiombymeansoftheotherusualassumptionsisnow
3、seldomundertaken,andthosewhodoundertakeareconsideredintheclasswithit,circle-squarersandsearchersforperpetualmotionsadby-productsofthecreativeactivityofmodernscience.Inthis,asinallotherchanges,thereissubjectbothforrejoicingandregret.Itisasatisfactiontoawrit
4、eronnon-euclideangeometrythathemayproceedatoncetohissubject,withoutfeelinganyneedtojustifyhimself,or,atleast,anymoreneedthananyotherwhoaddstooursupplyofbooks.Ontheotherhand,hewillmissthestimulusthatcomestoonewhofeelsthatheisbringingoutsomethingentirelynewa
5、ndstrange.Thesubjectofnon-euclideangeometryis,tothemathematician,quiteaswellestablishedasanyotherbranchofmathematicalscienceinfact,it;and,maylayclaimtoadecidedlymoresolidbasisthansomebranches,suchasthetheoryofassemblages,ortheanalysissitus.Recentbooksdeali
6、ngwithnon-euclideangeometryfallnaturallyintotwoclasses.IntheonewefindtheworksofKilling,Liebmann,andManning,*who*Detailedreferencesgivenlater.A24PREFACEwishtobuildupcertainclearlyconceivedgeometricalsystems,andarecarelessofthedetailsofthefounda-tionsonwhich
7、allistorest.IntheothercategoryareHilbert,Vahlen,Veronese,andtheauthorsofagoodlynumberofarticlesonthefoundationsofgeometry.Thesewritersdealatlengthwiththeconsistency,significance,andlogicalindependenceoftheirassumptions,butdonotgoveryfartowardsraisingasuper
8、structureonanyoneofthefoundationssuggested.Thepresentworkis,inameasure,anattempttounitethetwotendencies.Theauthor'sowninterest,beitstatedattheoutset,liesmainlyinthefruits,ratherthanintherootsbutthedayispastwh
