Euclid’s Plane Geometry

1. Euclid’s Plane Geometry By: Jamie Storm & Rebecca Krumrine

2. Preview • Babylonian “Geometry” • Egyptian “Geometry” • Thales’ contribution & Pythagoras’ Contribution • Plato’s contribution • Aristotle’s contribution • Euclidian Geometry

3. Babylonian “Geometry”(2000-500B.C.) • “Experimentally derived rules” used by engineers • Ancient clay tablets reveal that the Babylonian’s knew the Pythagorean relationship. Example: 4 is the length and 5 the diagonal. What is the breadth? It’s 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.

4. Egyptian “Geometry”(2000-500B.C.) • “Experimentally derived rules” used by engineers • The Egyptian Pyramid is evidence of their knowledge of Geometry

5. 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 a2+b2=c2

6. 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

7. 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.

8. ?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?

9. 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

10. Euclid’s Elements • Book 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.

11. 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.

12. 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

13. Euclid’s 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

14. Anyone know how to read Greek? GSP file

15. 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.

16. The Way of Thinking Euclid’s Elements show a person how to think logically about anything “The Elements is not just about shapes and numbers, it’s about how to think”

17. Who used this way of thinking? • French philosopher Rene Descartes • British Scientist Isaac Newton and Dutch Philosopher Baruch Spinoza • Early American Colonies • Abraham Lincoln

18. 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 Euclid’s logical structure.

19. Timeline • 4000-500BC Babylonian’s 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 Plato’s era; emphasized the idea of ‘proof’ and insisted on clear hypothesis • 384-232BC Aristotle’s era; introduces logical way of thinking • 300BC Euclid writes Elements

20. 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 Euclid’s 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

21. References • "A Short History of Geometry." SortSurfer.com. 2004. Unverstiry of St. Andrews, Scotland. 12 Nov 2006 <http://www.geometry algorithms.com/history.htm>. • 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 <http://en.wikipedia.org/wiki/Euclidean_geometry>. • Joyce , D. E.. "Book 1." Eucild's Elements. 1996. Clark University. 12 Nov 2006 <http://cs.clarku.edu/~djoyce/java/elements/bookI/bookI.html>. • 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 <http://math.rice.edu/~lanius/Geom/his.html>. • Morrow, Glenn R.. Proclus A Commentary on the First Book of Euclid's Elements. Princeton, New Jersey: Princeton University Press, 1970.