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TheProjectGutenbergEBookNon-EuclideanGeometry,byHenryManning
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Title:Non-EuclideanGeometry
Author:HenryManning
ReleaseDate:October10,2004[EBook#13702]
Language:English
Charactersetencoding:TeX
***STARTOFTHISPROJECTGUTENBERGEBOOKNON-EUCLIDEANGEOMETRY***
ProducedbyDavidStarner,JoshuaHutchinson,JohnHagerson,
andtheProjectGutenbergOn-lineDistributedProofreadingTeam.
i
NON-EUCLIDEAN
GEOMETRY
BY
HENRYPARKERMANNING,Ph.D.
AssistantProfessorofPureMathematics
inBrownUniversity
BOSTON,U.S.A.
GINN&COMPANY,PUBLISHERS.
T˙At˙numPre&
1901
Copyright,1901,by
HENRYPARKERMANNING
allrightsreserved
PREFACE
Non-EuclideanGeometryisnowrecognizedasanimportantbranchofMathe-
matics.ThosewhoteachGeometryshouldhavesomeknowledgeofthissubject,
andallwhoareinterestedinMathematicswillfindmuchtostimulatethemand
muchforthemtoenjoyinthenovelresultsandviewsthatitpresents.
ThisbookisanattempttogiveasimpleanddirectaccountoftheNon-
EuclideanGeometry,andonewhichpresupposesbutlittleknowledgeofMath-
ematics.ThefirstthreechaptersassumeaknowledgeofonlyPlaneandSolid
GeometryandTrigonometry,andtheentirebookcanbereadbyonewhohas
takenthemathematicalcoursescommonlygiveninourcolleges.
Nospecialclaimtooriginalitycanbemadeforwhatispublishedhere.The
propositionshavelongbeenestablished,andinvariousways.Someoftheproofs
maybenew,butothers,asalreadygivenbywritersonthissubject,couldnotbe
improved.ThesehavecometomechieflythroughthetranslationsofProfessor
GeorgeBruceHalstedoftheUniversityofTexas.
Iamparticularlyindebtedtomyfriend,ArnoldB.Chace,Sc.D.,ofValley
Falls,R.I.,withwhomIhavestudiedanddiscussedthesubject.
HENRYP.MANNING.
Providence,January,1901.
ii
Contents
PREFACE
ii
1INTRODUCTION
1
2PANGEOMETRY 3
2.1PropositionsDependingOnlyonthePrincipleofSuperposition. 3
2.2PropositionsWhichAreTrueforRestrictedFigures ....... 6
2.3TheThreeHypotheses........................ 9
3THEHYPERBOLICGEOMETRY 25
3.1ParallelLines............................. 25
3.2Boundary-curvesandSurfaces,andEquidistant-curvesandSurfaces35
3.3TrigonometricalFormulæ...................... 42
4THEELLIPTICGEOMETRY 51
5ANALYTICNON-EUCLIDEANGEOMETRY 56
5.1HyperbolicAnalyticGeometry................... 56
5.2EllipticAnalyticGeometry..................... 68
5.3EllipticSolidAnalyticGeometry.................. 74
6HISTORICALNOTE
79
7PROJECTGUTENBERG”SMALLPRINT”
iii
Chapter1
INTRODUCTION
TheaxiomsofGeometrywereformerlyregardedaslawsofthoughtwhichan
intelligentmindcouldneitherdenynorinvestigate.Notonlyweretheaxioms
towhichwehavebeenaccustomedfoundtoagreewithourexperience,butit
wasbelievedthatwecouldnotreasononthesuppositionthatanyofthemare
nottrue,ithasbeenshown,however,thatitispossibletotakeasetofaxioms,
whollyorinpartcontradictingthoseofEuclid,andbuildupaGeometryas
consistentashis.
WeshallgivethetwomostimportantNon-EuclideanGeometries. 1 Inthese
theaxiomsanddefinitionsaretakenasinEuclid,withtheexceptionofthose
relatingtoparallellines.Omittingtheaxiomonparallels, 2 weareledtothree
hypotheses;oneoftheseestablishestheGeometryofEuclid,whileeachofthe
othertwogivesusaseriesofpropositionsbothinterestinganduseful.Indeed,as
longaswecanexaminebutalimitedportionoftheuniverse,itisnotpossible
toprovethatthesystemofEuclidistrue,ratherthanoneofthetwoNon-
EuclideanGeometrieswhichweareabouttodescribe.
Weshalladoptanarrangementwhichenablesustoprovefirsttheproposi-
tionscommontothethreeGeometries,thentoproduceaseriesofpropositions
andthetrigonometricalformulæforeachofthetwoGeometrieswhichdier
fromthatofEuclid,andbyanalyticalmethodstoderivesomeoftheirmost
strikingproperties.
WedonotproposetoinvestigatedirectlythefoundationsofGeometry,nor
eventopointoutalloftheassumptionswhichhavebeenmade,consciouslyor
unconsciously,inthisstudy.LeavingundisturbedthatwhichtheseGeometries
haveincommon,wearefreetofixourattentionupontheirdierences.Bya
concreteexpositionitmaybepossibletolearnmoreofthenatureofGeometry
thanfromabstracttheoryalone.
1 SeeHistoricalNote,p.80.
2 Seep.79.
1

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