Euclid’s Plane Euclid’s Plane GeometryGeometry

By: Jamie StormBy: Jamie Storm

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Rebecca KrumrineRebecca Krumrine

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►Babylonian “Geometry”Babylonian “Geometry”►Egyptian “Geometry”Egyptian “Geometry”►Thales’ contribution & Pythagoras’ Thales’ contribution & Pythagoras’

ContributionContribution►Plato’s contributionPlato’s contribution►Aristotle’s contributionAristotle’s contribution►Euclidian GeometryEuclidian Geometry

Babylonian “Geometry”Babylonian “Geometry”(2000-500B.C.)(2000-500B.C.)

► ““Experimentally derived rules” used by engineersExperimentally derived rules” used by engineers► Ancient clay tablets reveal that the Babylonian’s Ancient clay tablets reveal that the Babylonian’s

knew the Pythagorean relationship.knew the Pythagorean relationship.

Example: 4 is the length and 5 the Example: 4 is the length and 5 the diagonal. What is the breadth? It’s size diagonal. What is the breadth? It’s size is not known.is not known.Solution: 4 times 4 is 16. 5 times 5 is 25. Solution: 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. You take 16 from 25 and there remains 9. What times what shall I take in order to get What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.9? 3 times 3 is 9. 3 is the breadth.

Egyptian “Geometry”Egyptian “Geometry”(2000-500B.C.)(2000-500B.C.)

►““Experimentally derived rules” used by Experimentally derived rules” used by engineersengineers

►The Egyptian Pyramid is evidence of The Egyptian Pyramid is evidence of their knowledge of Geometrytheir knowledge of Geometry

Paving the Way to Euclid…Paving the Way to Euclid…► ThalesThales

Greek historians refer to him as the father Greek historians refer to him as the father of geometryof geometry

Able to determine the height of a pyramid Able to determine the height of a pyramid by measuring the length of its shadow at by measuring the length of its shadow at a particular time of daya particular time of day

►PythagorasPythagorasProved that all the angles of a triangle Proved that all the angles of a triangle summed to the value of two right anglessummed to the value of two right anglesMost famous discovery was the Most famous discovery was the Pythagorean Theorem aPythagorean Theorem a22+b+b22=c=c22

Paving the way continuedPaving the way continued► Plato:Plato:

Above the entry door into his school, he wrote Above the entry door into his school, he wrote “Let No One Ignorant of Geometry Enter My “Let No One Ignorant of Geometry Enter My Doors”Doors”

Described two different methods towards the Described two different methods towards the development of Geometrydevelopment of Geometry

1) Start with a hypothesis and build upon this with 1) Start with a hypothesis and build upon this with the use of diagrams and images until you are able the use of diagrams and images until you are able to prove or disprove the hypothesis.to prove or disprove the hypothesis.2) Begin with a hypothesis and build upon that with 2) Begin with a hypothesis and build upon that with additional hypotheses until a principal is reached additional hypotheses until a principal is reached where there is nothing hypothetical. Then it is where there is nothing hypothetical. Then it is possible to descend back through all the previous possible to descend back through all the previous steps and prove the original hypothesis.steps and prove the original hypothesis.Emphasized the idea of proof, and insisted on Emphasized the idea of proof, and insisted on accurate definitions and clear hypothesesaccurate definitions and clear hypotheses

Paving the Way ContinuedPaving the Way Continued

►AristotleAristotle Pointed out that a logical system Pointed out that a logical system

must begin with a few basic must begin with a few basic assumptions to build upon.assumptions to build upon.

Logical argument was the only Logical argument was the only certain way of obtaining scientific certain way of obtaining scientific knowledge.knowledge.

?What is Geometry??What is Geometry?

► If you were developing Geometry, how If you were developing Geometry, how would you start?would you start?

►What do you think are the most What do you think are the most important definitions of plane important definitions of plane Euclidean geometry?Euclidean geometry?

EuclidEuclid►Used what was known, as well as his Used what was known, as well as his

own work to develop 465 propositionsown work to develop 465 propositions►13 books 13 books ElementsElements

plane and solid geometryplane and solid geometry algebraalgebra trigonometrytrigonometry advanced arithmeticadvanced arithmetic

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

Euclid’s Euclid’s ElementsElements

►Book 1 DefinitionsBook 1 Definitions

Note: It is important to realize that these definitions were not Euclid’s original ideas. His book however was the first work to contain these definition and survive time.

10 basic assumptions10 basic assumptions► These are considered the starting points of These are considered the starting points of

geometry and do not require proofgeometry and do not require proof► PostulatesPostulates

A straight line can be drawn from any point to A straight line can be drawn from any point to any pointany point

A finite straight line can be extended A finite straight line can be extended continuously in a straight line.continuously in a straight line.

A circle can be formed with any center and A circle can be formed with any center and distance (radius)distance (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 If a straight line falling on two straight lines

makes the sum of the interior angles on the makes the sum of the interior angles on the same side less than two right angles, then the same side less than two right angles, then the two straight lines, if extended indefinitely , meet two straight lines, if extended indefinitely , meet on the side on which the angle sum is less than on the side on which the angle sum is less than the two right angles. the two right angles.

10 basic assumptions10 basic assumptions►5 common notations5 common notations

Things equal to the same thing are also Things equal to the same thing are also equal to each otherequal to each otherIf equals are added to equals, the results If equals are added to equals, the results are equalare equalIf equals are subtracted from equals, the If equals are subtracted from equals, the remainders are equalremainders are equalThings that coincide with one another are Things that coincide with one another are equal to one anotherequal to one anotherThe whole is greater than the partThe whole is greater than the part

Euclid’s First ProofEuclid’s First Proof

►Prove that you can construct an equilateral Prove that you can construct an equilateral triangle from a finite straight line.triangle from a finite straight line.

►Given: Let AB be the given finite straight Given: Let AB be the given finite straight line.line.

►Hint: This involves the construction of Hint: This involves the construction of circlescircles

GSP file

Anyone know how to read Greek?

Additional ProofsAdditional Proofs

►Two triangles are congruentTwo triangles are congruent► Isosceles Triangle TheoremIsosceles Triangle Theorem► If two triangle angles equal one-If two triangle angles equal one-

another, then the sides opposite one another, then the sides opposite one another equal one anotheranother equal one another

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

The Way of ThinkingThe Way of Thinking

Euclid’s Elements show a person how Euclid’s Elements show a person how to think logically about anythingto think logically about anything

“ “The The ElementsElements is not just is not just

about shapes and numbers, about shapes and numbers,

it’s about how to think”it’s about how to think”

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

► French philosopher Rene DescartesFrench philosopher Rene Descartes

►British Scientist Isaac Newton and British Scientist Isaac Newton and Dutch Philosopher Baruch SpinozaDutch Philosopher Baruch Spinoza

►Early American ColoniesEarly American Colonies

►Abraham LincolnAbraham Lincoln

TodayToday

► In the 20In the 20thth Century, the study of Century, the study of Geometry migrated from the Geometry migrated from the Universities to the High Schools.Universities to the High Schools.

►The two-column proof made it easier The two-column proof made it easier for students to understand.for students to understand.

►There is a de-emphasis on Euclid’s There is a de-emphasis on Euclid’s logical structure. logical structure.

TimelineTimeline► 4000-500BC Babylonian’s had experimentally derived 4000-500BC Babylonian’s had experimentally derived

relationships & they also solved Pythagorean relationships & they also solved Pythagorean relationships on clay tablesrelationships on clay tables

► 2000-500BC Egyptian engineers used experimentally 2000-500BC Egyptian engineers used experimentally derived rules derived rules

► 625-547BC Thales era; contributed practical applications 625-547BC Thales era; contributed practical applications of geometryof geometry

► 569-475BC Pythagoras era; contributed his ideas 569-475BC Pythagoras era; contributed his ideas including the Pythagorean theoremincluding the Pythagorean theorem

► 427-347BC Plato’s era; emphasized the idea of ‘proof’ and 427-347BC Plato’s era; emphasized the idea of ‘proof’ and insisted on clear hypothesisinsisted on clear hypothesis

► 384-232BC Aristotle’s era; introduces logical way of 384-232BC Aristotle’s era; introduces logical way of thinkingthinking

► 300BC Euclid writes300BC Euclid writes Elements Elements

TimelineTimeline

► 1717thth C. Rene Descartes bases part of his philosophical C. Rene Descartes bases part of his philosophical method on the “long chains of reasoning”method on the “long chains of reasoning”

► 1717thth C. Isaac Newton and Baruch Spinoza used the C. Isaac Newton and Baruch Spinoza used the form of Euclid’s form of Euclid’s ElementsElements to present their ideas to present their ideas

► 1818thth C The 13 American colonies broke away from C The 13 American colonies broke away from Great Britain by agreeing to the Declaration of Great Britain by agreeing to the Declaration of IndependenceIndependence

► 1919thth C. Abraham Lincoln carried a copy of C. Abraham Lincoln carried a copy of ElementsElements with him and studied it with him and studied it

► 2020thth C. The study of Geometry begins to be taught in C. The study of Geometry begins to be taught in high schoolshigh schools

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 <http://www.geometry Andrews, Scotland. 12 Nov 2006 <http://www.geometry algorithms.com/history.htm>.algorithms.com/history.htm>.

► Berlinghoff, William P. , and Fernando Q. Gouvea. Berlinghoff, William P. , and Fernando Q. Gouvea. Math through the Math through the Ages A Ages A Gentle History for Teachers and OthersGentle History for Teachers and Others. 1. 1stst ed. Farmington, ed. Farmington, Maine: Oxton Maine: Oxton House Publishers, 2002.House Publishers, 2002.

► Euclid. Euclid. Elements.Elements. Trans. with commentary by Sir Thomas L. Hearth. 2 Trans. 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 <http://en.wikipedia.org/wiki/Euclidean_geometry>.<http://en.wikipedia.org/wiki/Euclidean_geometry>.

► Joyce , D. E.. "Book 1." Joyce , D. E.. "Book 1." Eucild's ElementsEucild's Elements. 1996. Clark University. 12 Nov . 1996. Clark University. 12 Nov 2006 2006 <http://cs.clarku.edu/~djoyce/java/elements/bookI/bookI.html>.<http://cs.clarku.edu/~djoyce/java/elements/bookI/bookI.html>.

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

► Lanius, Cynthia. "History of Geometry." Lanius, Cynthia. "History of Geometry." Cynthia Lanius' LessonsCynthia Lanius' Lessons. 2004. . 2004. Rice Rice Univeristy. 12 Nov 2006 Univeristy. 12 Nov 2006 <http://math.rice.edu/~lanius/Geom/his.html>.<http://math.rice.edu/~lanius/Geom/his.html>.

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