03. euclid's elements (~300 b.c.) 1. 5 postulates 2. basic

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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