Transcript Euclid`s Plane Geometry

Euclid’s Plane Geometry
By: Jamie Storm
&
Rebecca Krumrine
Preview
► Babylonian
“Geometry”
► Egyptian “Geometry”
► Thales’ contribution & Pythagoras’
Contribution
► Plato’s contribution
► Aristotle’s contribution
► Euclidian Geometry
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.
Egyptian “Geometry”
(2000-500B.C.)
► “Experimentally
derived rules” used by
engineers
► The Egyptian Pyramid is evidence of their
knowledge of Geometry
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
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
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.
?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?
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
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.
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.
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
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
Anyone know how to read Greek?
GSP file
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.
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”
Who used this way of thinking?
►
French philosopher Rene Descartes
►British
Scientist Isaac Newton and Dutch
Philosopher Baruch Spinoza
►Early
American Colonies
►Abraham
Lincoln
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.
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
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
References
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"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.