03.Euclid'sElements (~300B.C.)
~100A.D.Earliestexistingcopy
1570A.D.FirstEnglishtranslation
1956DoverEdition
• Contents:I. DefinitionsII. PostulatesIII. CommonNotionsIV. Propositions
• Euclid'sAccomplishment: Showedthatallgeometricclaimsthenknownfollowfrom5postulates.
basicassumptions
morecomplexclaims⇒
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1.5Postulates2.BasicConcepts3.EuclideanGeometryasaTheoryofSpace
1.Euclid's5Postulates(i) Todrawastraightlinefromanypointtoanypoint.
(ii) Toproduceafinitestraightlinecontinuouslyinastraightline.
(iii) Todescribeacirclewithanycenteranddistance.
(iv) Thatallrightangles areequaltooneanother.
•
A B• •
A B• •
Def.10.Whenastraightlinesetuponastraightlinemakestheadjacentanglesequaltooneanother,eachoftheequalanglesisright...
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(v) That,ifastraightlinefallingontwostraightlinesmakestheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhichtheanglesarelessthantworightangles.
•
α+β<90o
α
β
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2.Basicconcepts(i) "point"="thatwhichhasnopart"
- Moderngloss:nomagnitude,nodimension.
(ii) "line"="breadthlesslength"- Moderngloss:only length;i.e.,1-dimensional.
Twomoderncharacteristicsofaline.
(b)Alineisastraight curve.
(a)Alineisadensecollectionofpointswithnogaps.
• Densecollectionofpoints =betweenanytwopointsisanother.• Densenessdoesnot imply"nogaps":- Thecollectionofrationalnumbers(ratiosofnaturalnumbers)isdensebutcontainsgaps(theirrationalnumbers).
- Thecollectionofrealnumbers(rationalsandirrationals)isdenseandcontainsno gaps.
• Adense collectionofpointswithnogaps definesacontinuum.
• Astraightcurveisacurvesuchthatthedistancebetweenanyofitspointsistheshortest.
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3.EuclideanGeometryasaTheoryofSpace• Recall:TwoquestionstoaskofatheoryT:(i) IsTconsistent? (Ifyes,thentherearepossibleworldsin
whichT istrue.)(ii) DoesTaccuratelydescribetheactualworld?
• Euclideangeometryis consistent(Hilbert1899).
• But:Doesitacuratelydescribetheactualworld?- Consideritspropositionsaspredictions aboutthepropertiesofspace
• So:IfPropositionx isfalseintheactualworld,thenoneormoreofthepremisesmustbefalseintheactualworld.
1.Postulates1-5.----------------------\ Propositionx.
- BecauseEuclideangeometryisconsistent,eachpropositionistheconclusionofavalid-deductive argument,ultimatelyoftheform:
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Examplesofpropositions
Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequal,theexteriorangleequaltotheinteriorandoppositeangle,theinterioranglesonthesamesideequaltotworightangles.
A B
C D
E
F
G
H
Claims:(a) !AGH =!GHD.(b) !EGB =!GHD.(c) !BGH +!GHD=tworightangles.
Proposition13.Ifastraightlinestandsonastraightline,thenitmakeseithertworightangles,orangleswhosesumequalstworightangles.
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Proposition32.Inanytriangle,ifoneofthesidesisextended,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.
A
B C D
Claims:(a) !ACD=!CAB+!ABC.(b) !CAB+!ABC+!BCA=tworightangles.
Proof:1. DrawCE paralleltoBA. (Prop.31)
A
B C D
E
2. !CAB=!ACE. (Prop.29:alternateanglesareequal.)
3. !ECD=!ABC. (Prop.29:exteriorangle =interioroppositeangle)
5. !ACD+!BCA=!CAB+!ABC+!BCA (CommonNotion2)
6. !ACD+!BCA=tworightangles (Prop.13)
7. !CAB+!ABC +!BCA=tworightangles (CommonNotion1:Thingsequaltothesamethingareequaltoeachother.)
4. !ACD =!ACE+!ECD=!CAB+!ABC
(CommonNotion2:Ifequalsareaddedtoequalsthentheresultingwholesareequal.)
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Now:IsProp.32trueintheactualworld?• Suppose:WeconstructamassivetrianglebetweenthreemountainpeaksinBavaria,andmeasureitsanglesusingreflectedlightrays.
• Considerthefollowingvalid-deductive argument:
1. Lightraystravelalongstraightlines.2. Euclideangeometry(i.e.,Postulates1-5)istrueintheactualworld.-------------------------------------------------------------------\ ThesumoftheanglesoftheBavariantriangle=tworightangles.
• Suppose:Theconclusionisfalse.• Then:Oneormorepremisesmustbefalse...butwhichones?
- Ifthesumoftheangles>180°,thenperhapslightrays"bulgeoutwards"betweenmountainpeaks.
- Ifthesumoftheangles<180°,thenperhapslightrays"bulgeinwards"betweenmountainpeaks.
• Canupholdpremise2bydenyingpremise1:
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Now:IsProp.32trueintheactualworld?• Suppose:WeconstructamassivetrianglebetweenthreemountainpeaksinBavaria,andmeasureitsanglesusingreflectedlightrays.
• Considerthefollowingvalid-deductive argument:
• Suppose:Theconclusionisfalse.• Then:Oneormorepremisesmustbefalse...butwhichones?
- Ifthesumoftheangles>180°,thenperhapsspherical geometryistrueintheactualworld.
- Ifthesumoftheangles<180°,thenperhapshyperbolic geometryistrueintheactualworld.
• Canupholdpremise1bydenyingpremise2:
1. Lightraystravelalongstraightlines.2. Euclideangeometry(i.e.,Postulates1-5)istrueintheactualworld.-------------------------------------------------------------------\ ThesumoftheanglesoftheBavariantriangle=tworightangles.
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