Euclid

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Euclids Plane Geometry

• By Jamie Storm
• Rebecca Krumrine

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Preview

• Babylonian Geometry
• Egyptian Geometry
• Thales contribution Pythagoras Contribution
• Platos contribution
• Aristotles contribution
• Euclidian Geometry

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Babylonian Geometry(2000-500B.C.)

• Experimentally derived rules used by engineers
• Ancient clay tablets reveal that the Babylonians
  knew the Pythagorean relationship.

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

• Solution 4 times 4 is 16. 5 times 5 is 25. You
  take 16 from 25 and there remains 9. What times
  what shall I take in order to get 9? 3 times 3
  is 9. 3 is the breadth.

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Egyptian Geometry(2000-500B.C.)

• Experimentally derived rules used by engineers
• The Egyptian Pyramid is evidence of their
  knowledge of Geometry

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Paving the Way to Euclid

• Thales
• Greek historians refer to him as the father of
  geometry
• Able to determine the height of a pyramid by
  measuring the length of its shadow at a
  particular time of day

• Pythagoras
• Proved that all the angles of a triangle summed
  to the value of two right angles
• Most famous discovery was the Pythagorean Theorem
  a2b2c2

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Paving the way continued

• Plato
• Above the entry door into his school, he wrote
  Let No One Ignorant of Geometry Enter My Doors
• Described two different methods towards the
  development of Geometry

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

• Aristotle
• Pointed out that a logical system must begin with
  a few basic assumptions to build upon.
• Logical argument was the only certain way of
  obtaining scientific knowledge.

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?What is Geometry?

• If you were developing Geometry, how would you
  start?
• What do you think are the most important
  definitions of plane Euclidean geometry?

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Euclid

• Used what was known, as well as his own work to
  develop 465 propositions
• 13 books Elements
• plane and solid geometry
• algebra
• trigonometry
• advanced arithmetic

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

• Book 1 Definitions

Note It is important to realize that these
definitions were not Euclids original ideas.
His book however was the first work to contain
these definition and survive time.
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10 basic assumptions

• These are considered the starting points of
  geometry and do not require proof
• Postulates
• A straight line can be drawn from any point to
  any point
• A finite straight line can be extended
  continuously in a straight line.
• A circle can be formed with any center and
  distance (radius)
• All right angles are equal to one another.
• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than two right angles, then the two
  straight lines, if extended indefinitely , meet
  on the side on which the angle sum is less than
  the two right angles.

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10 basic assumptions

• 5 common notations
• Things equal to the same thing are also equal to
  each other
• If equals are added to equals, the results are
  equal
• If equals are subtracted from equals, the
  remainders are equal
• Things that coincide with one another are equal
  to one another
• The whole is greater than the part

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Euclids First Proof

• Prove that you can construct an equilateral
  triangle from a finite straight line.
• Given Let AB be the given finite straight line.

• Hint This involves the construction of circles

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Anyone know how to read Greek?
GSP file
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Additional Proofs

• Two triangles are congruent
• Isosceles Triangle Theorem
• If two triangle angles equal one-another, then
  the sides opposite one another equal one another
• Basic constructions of midpoints of lines,
  perpendicular lines etc.

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The Way of Thinking

• Euclids Elements show a person how to think
  logically about anything
• The Elements is not just
• about shapes and numbers,
• its about how to think

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Who used this way of thinking?

• French philosopher Rene Descartes

• British Scientist Isaac Newton and Dutch
  Philosopher Baruch Spinoza

• Early American Colonies

• Abraham Lincoln

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Today

• In the 20th Century, the study of Geometry
  migrated from the Universities to the High
  Schools.
• The two-column proof made it easier for students
  to understand.
• There is a de-emphasis on Euclids logical
  structure.

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Timeline

• 4000-500BC Babylonians had experimentally
  derived relationships they also solved
  Pythagorean relationships on clay tables
• 2000-500BC Egyptian engineers used experimentally
  derived rules
• 625-547BC Thales era contributed practical
  applications of geometry
• 569-475BC Pythagoras era contributed his ideas
  including the Pythagorean theorem
• 427-347BC Platos era emphasized the idea of
  proof and insisted on clear hypothesis
• 384-232BC Aristotles era introduces logical way
  of thinking
• 300BC Euclid writes Elements

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Timeline

• 17th C. Rene Descartes bases part of his
  philosophical method on the long chains of
  reasoning
• 17th C. Isaac Newton and Baruch Spinoza used the
  form of Euclids Elements to present their ideas
• 18th C The 13 American colonies broke away from
  Great Britain by agreeing to the Declaration of
  Independence
• 19th C. Abraham Lincoln carried a copy of
  Elements with him and studied it
• 20th C. The study of Geometry begins to be taught
  in high schools

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References

• "A Short History of Geometry." SortSurfer.com.
  2004. Unverstiry of St. Andrews, Scotland. 12
  Nov 2006 lthttp//www.geometry algorithms.com/his
  tory.htmgt.
• Berlinghoff, William P. , and Fernando Q. Gouvea.
  Math through the Ages A Gentle History for
  Teachers and Others. 1st ed. Farmington, Maine
  Oxton House Publishers, 2002.
• Euclid. Elements. Trans. with commentary by Sir
  Thomas L. Hearth. 2nd ed. New York Dover
  Publications, 1956. 
• "Euclidean Geometry." Wikipedia. 2006. Wikipedia
  . 12 Nov 2006 lthttp//en.wikipedia.org/wiki/Eucli
  dean_geometrygt.
• Joyce , D. E.. "Book 1." Eucild's Elements. 1996.
  Clark University. 12 Nov 2006 lthttp//cs.clarku.e
  du/djoyce/java/elements/bookI/bookI.htmlgt.
• Katz, Victor J.. A History of Mathematics. New
  York Pearson/Addison-Wesley, 2004.
• Lanius, Cynthia. "History of Geometry." Cynthia
  Lanius' Lessons. 2004. Rice Univeristy. 12 Nov
  2006 lthttp//math.rice.edu/lanius/Geom/his.htmlgt.
  
• Morrow, Glenn R.. Proclus A Commentary on the
  First Book of Euclid's Elements. Princeton, New
  Jersey Princeton University Press, 1970.