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    Euclids Plane GeometryEuclids Plane Geometry

    By: Jamie StormBy: Jamie Storm

    &&

    Rebecca KrumrineRebecca Krumrine

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    PreviewPreview

    Babylonian GeometryBabylonian Geometry

    Egyptian GeometryEgyptian Geometry

    Thales contribution & PythagorasThales contribution & PythagorasContributionContribution

    Platos contributionPlatos contribution

    Aristotles contributionAristotles contributionEuclidian GeometryEuclidian Geometry

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    Babylonian GeometryBabylonian Geometry

    (2000(2000--500B.C.)500B.C.)Experimentally derived rules used by engineersExperimentally derived rules used by engineers

    Ancient clay tablets reveal that the BabyloniansAncient clay tablets reveal that the Babyloniansknew the Pythagorean relationship.knew the Pythagorean relationship.

    Example: 4 is the length and 5 the diagonal.Example: 4 is the length and 5 the diagonal.What is the breadth? Its size is not known.What is the breadth? Its size is not known.

    Solution: 4 times 4 is 16. 5 times 5 is 25. YouSolution: 4 times 4 is 16. 5 times 5 is 25. Youtake 16 from 25 and there remains 9. What timestake 16 from 25 and there remains 9. What timeswhat shall I take in order to get 9? 3 times 3 is 9.what shall I take in order to get 9? 3 times 3 is 9.3 is the breadth.3 is the breadth.

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    Paving the Way to EuclidPaving the Way to Euclid

    ThalesThales

    Greek historians refer to him as the father ofGreek historians refer to him as the father ofgeometrygeometry

    Able to determine the height of a pyramid byAble to determine the height of a pyramid bymeasuring the length of its shadow at ameasuring the length of its shadow at aparticular time of dayparticular time of day

    PythagorasPythagoras

    Proved that all the angles of a triangle summedProved that all the angles of a triangle summedto the value of two right anglesto the value of two right angles

    Most famous discovery was the PythagoreanMost famous discovery was the Pythagorean

    Theorem aTheorem a22

    +b+b22

    =c=c22

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    Paving the way continuedPaving the way continued Plato:Plato:

    Above the entry door into his school, he wrote Let NoAbove the entry door into his school, he wrote Let NoOne Ignorant of Geometry Enter My DoorsOne Ignorant of Geometry Enter My Doors

    Described two different methods towards theDescribed two different methods towards thedevelopment of Geometrydevelopment of Geometry

    1) Start with a hypothesis and build upon this with the use1) Start with a hypothesis and build upon this with the useof diagrams and images until you are able to prove orof diagrams and images until you are able to prove ordisprove the hypothesis.disprove the hypothesis.2) Begin with a hypothesis and build upon that with2) Begin with a hypothesis and build upon that with

    additional hypotheses until a principal is reached whereadditional hypotheses until a principal is reached wherethere is nothing hypothetical. Then it is possible tothere is nothing hypothetical. Then it is possible todescend back through all the previous steps and prove thedescend back through all the previous steps and prove theoriginal hypothesis.original hypothesis.Emphasized the idea of proof, and insisted on accurateEmphasized the idea of proof, and insisted on accurate

    definitions and clear hypothesesdefinitions and clear hypotheses

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    Paving the Way ContinuedPaving the Way Continued

    AristotleAristotle

    Pointed out that a logical system mustPointed out that a logical system must

    begin with a few basic assumptions tobegin with a few basic assumptions tobuild upon.build upon.

    Logical argument was the only certainLogical argument was the only certainway of obtaining scientific knowledge.way of obtaining scientific knowledge.

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    ?What is Geometry??What is Geometry?

    If you were developing Geometry, howIf you were developing Geometry, howwould you start?would you start?

    What do you think are the most importantWhat do you think are the most importantdefinitions of plane Euclidean geometry?definitions of plane Euclidean geometry?

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    EuclidEuclid

    Used what was known, as well as his ownUsed what was known, as well as his ownwork to develop 465 propositionswork to develop 465 propositions

    13 books13 books ElementsElements

    plane and solid geometryplane and solid geometry

    algebraalgebra

    trigonometrytrigonometry

    advanced arithmeticadvanced arithmetic

    -no other book except the Bible has been circulatedmore widely throughout the world, more edited ormore studied

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    EuclidsEuclids ElementsElements

    Book 1 DefinitionsBook 1 Definitions

    Note: It is important to realize that these definitionswere not Euclids original ideas. His book howeverwas the first work to contain these definition andsurvive time.

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    10 basic assumptions10 basic assumptions These are considered the starting points of geometry andThese are considered the starting points of geometry and

    do not require proofdo not require proof PostulatesPostulates

    A straight line can be drawn from any point to any pointA straight line can be drawn from any point to any point A finite straight line can be extended continuously in aA finite straight line can be extended continuously in a

    straight line.straight line. A circle can be formed with any center and distanceA circle can be formed with any center and distance

    (radius)(radius) All right angles are equal to one another.All right angles are equal to one another. If a straight line falling on two straight lines makes theIf a straight line falling on two straight lines makes the

    sum of the interior angles on the same side less thansum of the interior angles on the same side less thantwo right angles, then the two straight lines, if extendedtwo right angles, then the two straight lines, if extendedindefinitely , meet on the side on which the angle sumindefinitely , meet on the side on which the angle sumis less than the two right angles.is less than the two right angles.

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    10 basic assumptions10 basic assumptions

    5 common notations5 common notations

    Things equal to the same thing are also equal toThings equal to the same thing are also equal toeach othereach other

    If equals are added to equals, the results areIf equals are added to equals, the results areequalequal

    If equals are subtracted from equals, theIf equals are subtracted from equals, theremainders are equalremainders are equal

    Things that coincide with one another are equalThings that coincide with one another are equalto one anotherto one another

    The whole is greater than the partThe whole is greater than the part

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    Euclids First ProofEuclids First Proof

    Prove that you can construct an equilateralProve that you can construct an equilateraltriangle from a finite straight line.triangle from a finite straight line.

    Given: Let AB be the given finite straightGiven: Let AB be the given finite straightline.line.

    Hint: This involves the construction of circlesHint: This involves the construction of circles

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    GSP file

    Anyone know how to read Greek?

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    Additional ProofsAdditional Proofs

    Two triangles are congruentTwo triangles are congruent

    Isosceles Triangle TheoremIsosceles Triangle Theorem

    If two triangle angles equal oneIf two triangle angles equal one--another,another,then the sides opposite one another equalthen the sides opposite one another equalone anotherone another

    Basic constructions of midpoints of lines,Basic constructions of midpoints of lines,perpendicular lines etc.perpendicular lines etc.

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    The Way of ThinkingThe Way of Thinking

    Euclids Elements show a person how toEuclids Elements show a person how tothink logically about anythingthink logically about anything

    TheThe ElementsElements is not justis not just

    about shapes and numbers,about shapes and numbers,

    its about how to thinkits about how to think

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    Who used this way of thinking?Who used this way of thinking?

    French philosopher Rene DescartesFrench philosopher Rene Descartes

    British Scientist Isaac Newton and DutchBritish Scientist Isaac Newton and DutchPhilosopher Baruch SpinozaPhilosopher Baruch Spinoza

    Early American ColoniesEarly American Colonies

    Abraham LincolnAbraham Lincoln

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    TodayToday

    In the 20In the 20thth Century, the study of GeometryCentury, the study of Geometrymigrated from the Universities to the Highmigrated from the Universities to the HighSchools.Schools.

    The twoThe two--column proof made it easier forcolumn proof made it easier forstudents to understand.students to understand.

    There is a deThere is a de--emphasis on Euclids logicalemphasis on Euclids logicalstructure.structure.

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    TimelineTimeline 40004000--500BC Babylonians had experimentally derived500BC Babylonians had experimentally derived

    relationships & they also solved Pythagorean relationshipsrelationships & they also solved Pythagorean relationshipson clay tableson clay tables

    20002000--500BC Egyptian engineers used experimentally500BC Egyptian engineers used experimentallyderived rulesderived rules

    625625--547BC Thales era; contributed practical applications of547BC Thales era; contributed practical applications ofgeometrygeometry

    569569--475BC Pythagoras era; contributed his ideas including475BC Pythagoras era; contributed his ideas includingthe Pythagorean theoremthe Pythagorean theorem

    427427--347BC Platos era; emphasized the idea of proof and347BC Platos era; emphasized the idea of proof andinsisted on clear hypothesisinsisted on clear hypothesis

    384384--232BC Aristotles era; introduces logical way of232BC Aristotles era; introduces logical way ofthinkingthinking

    300BC Euclid writes300BC Euclid writes ElementsElements

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    ReferencesReferences

    "A Short History of Geometry.""A Short History of Geometry." SortSurfer.comSortSurfer.com. 2004. Unverstiry of St.. 2004. Unverstiry of St.Andrews, Scotland. 12 Nov 2006 .

    Berlinghoff, William P. , and Fernando Q. Gouvea.Berlinghoff, William P. , and Fernando Q. Gouvea. Math through the Ages AMath through the Ages AGentle History for Teachers and OthersGentle History for Teachers and Others. 1. 1ststed. Farmington, Maine: Oxtoned. Farmington, Maine: OxtonHouse Publishers, 2002.House Publishers, 2002.

    Euclid.Euclid. Elements.Elements. Trans. with commentary by Sir Thomas L. Hearth. 2Trans. with commentary by Sir Thomas L. Hearth. 2ndnd ed.ed.New York: Dover Publications, 1956.New York: Dover Publications, 1956.

    "Euclidean Geometry.""Euclidean Geometry." WikipediaWikipedia. 2006. Wikipedia . 12 Nov 2006. 2006. Wikipedia . 12 Nov 2006..

    Joyce , D. E.. "Book 1."Joyce , D. E.. "Book 1." Eucild's ElementsEucild's Elements. 1996. Clark University. 12 Nov 2006. 1996. Clark University. 12 Nov 2006..

    Katz, Victor J..Katz, Victor J.. A History of Mathematics.A History of Mathematics. New York: Pearson/AddisonNew York: Pearson/Addison--Wesley,Wesley,2004.2004.

    Lanius, Cynthia. "History of Geometry."Lanius, Cynthia. "History of Geometry." Cynthia Lanius' LessonsCynthia Lanius' Lessons. 2004. Rice. 2004. RiceUniveristy. 12 Nov 2006 .Univeristy. 12 Nov 2006 .

    Morrow, Glenn R..Morrow, Glenn R.. Proclus A Commentary on the First Book of Euclid'sProclus A Commentary on the First Book of Euclid'sElementsElements. Princeton, New Jersey: Princeton University Press, 1970.. Princeton, New Jersey: Princeton University Press, 1970.