Euclid

1
Euclids Plane Geometry

• The Elements

2
Euclid 300s BCE

• Teacher at Museum and Library in Alexandria,
  founded by Ptolemy in 300 BCE.
• Best known for compiling and organizing the work
  of other Greek mathematicians relating to
  Geometry.

3
Aristotle 384-322 BCE

• Begin your scientific work with definitions
  and axioms.

4
The Elements

• Consisted of 13 volumes of definitions, axioms,
  theorems and proofs.
• Compilation of knowledge.
• The Elements was first math book in which each
  theorem was proved using axioms and previously
  proven theorems teaching how to think and
  develop logical arguments.
• Second only to the Bible in publications.

5

• Books 1-6 Plane Geometry
• 1-2 triangles, quadrilaterals, quadratics
• 3 - circles
• 4 - inscribed and circumscribed polygons
• 5 magnitudes and ratio, Euclidean Algorithm
• 6 applications of books 1-5
• Books 7-9 Number Theory
• Book 10 Irrational Numbers
• Books 11-13 Three dimensional figures
  including 5 Platonic solids

6
Book 1

• 5 statements that Euclid believed were obvious.
• 5 postulates about Geometry that Euclid believed
  were intuitively true.
• 23 definitions to help clarify the postulates
  (point, line, plane, angle etc)

7

• 5 Common notions (obvious)
• Things equal to the same thing are equal.
• If equals are added to equals, the results are
  equal.
• If equals are subtracted to equals, the results
  are equal.
• Things that coincide are equal.
• The whole is greater than the part.

8

• 5 assumptions (intuitively true)
• Postulate 1 a straight line can be drawn from
  any point to any point.
• (assumes only one line)
• Postulate 2 a line segment can be extended into
  a line.
• Postulate 3 a circle can be formed with any
  center and any radius
• (assumes only one circle)
• Postulate 4 all right angles are congruent
• Postulate 5 if two lines are cut by a
  transversal and the consecutive interior
  angles are not supplementary then the lines
  intersect.

9
Book I

• Included theorems such as
• Parallel Line Postulate
• Pythagorean Theorem
• construction of a square (using only a straight
  edge and protractor)
• SAS
• properties of parallelograms
• properties of parallel lines cut by a transversal

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Inscribed Polygons(Book IV)

• Euclid proved many theorems about circles in Book
  III that allowed him to provide detailed
  constructions of inscribed and circumscribed
  polygons.
• For example, to inscribe a pentagon, draw an
  isosceles triangle with the base angles equal to
  twice the vertex angle. Bisect the base angles
  and the 5 points together make the pentagon.

11
Duplicate Ratio(Book V)

• Book VII begins with a definition of proportional
  which is based on the notion of duplicate ratio.
• Duplicate ratio
• When three magnitudes are proportional, the
  first is said to have to the third the duplicate
  ratio of that which it has to the second.
• Ex. 2618

12
Euclidean Algorithm(Book VII)

• Process for finding the greatest common divisor.
• Given a, b with a gt b, subtract b from a
  repeatedly until get remainder c.
• Then subtract c from b repeatedly until get to m,
  then subtract m from cwhen the result 0, you
  have the greatest common divisor or the result
  1, which means a and b are relatively prime.
• ex. 80 and 18
• ex. 7 and 32

13
Prime Numbers

• Consider these 3 statements about primes found in
  Book VII
• Any composite number can be divided by some
  prime number.
• Any number is either prime or can be divided by
  a prime number.
• If a prime number can be divided into the
  product of two numbers, it can be divided into
  one of them.
• These statements form the Fundamental Theorem of
  Arithmetic that any number can be expressed
  uniquely as a product of prime numbers.
• In Book IX, Euclid proves through induction that
  there are infinitely many prime numbers.

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Geometric Series(Book IX)

• If as many as we please are in continued
  proportion, and there is subtracted from the
  second and the last numbers equal to the first,
  then, as the excess of the second is to the
  first, so will the last be to all those before
  it.

• a, ar, ar², ar³,...arn
• (ar a) (arn-a)
• (ar a)a (arn-a)Sn

15

• Solve this last equation for Sn
• (ar a)a (arn-a)Sn
• Ex. Find the sum of the first 5 terms when a 1
  and r 2

16
Knowing how to think- who needs it?

• Lawyers, politicians, negotiators, programmers,
  and anyone dealing with social issues!
• Abraham Lincoln carried a copy of The Elements
  (and read it) to become a better lawyer.
• The Declaration of Independence is set up in the
  same format as The Elements (self-evident truths
  are axioms used to prove that the colonies are
  justified in breaking from England).
• 19th century Yale students studied The Elements
  for two years, at the end of which they
  participated in a celebration ritual called the
  Burial of Euclid.
• E.T. Bell wrote Euclid taught me that without
  assumptions, there is no proof. Therefore, in any
  argument, examine the assumptions.

17
High School Geometry

• Plane Geometry courses today are basically the
  content of Euclids Elements.
• Two-column proof appeared in the 1900s to make
  proofs easier but led to rote memorization
  instead.
• 1970s moved away from proofs because they were
  too painful and not fun.
• Now proofs are brief and irrelevant. They do not
  serve the purpose of developing logical thinking.

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PSSA

• Standards what they should know
• Anchors what they are tested on

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Timeline

• Prior to Euclid, Greek mathematicians such as
  Pythagorus, Theaetetus, Euxodus and Thales did
  work in Geometry.
• 384-322 BCE - Aristotle believed that scientific
  knowledge could only be gained through logical
  methods, beginning with axioms.
• 300 BCE- Euclid teaches at the Museum and Library
  at Alexandria
• 1880 J.L. Heiberg compiles Greek version of The
  Elements as close to original as possible.
• 1908 Thomas Heath translated Heibergs text.
  This version is the one most widely used and the
  basis for modern Geometry courses.

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References

• Berlinghoff, F. Gouvêa. Math Through the Ages
  A Gentle History for Teachers and Others.
  Farmington, Maine Oxton House, 2002.
• Heath, T. History of Greek Mathematics, Volume 2.
  New York, 1981.
• Katz, V. The History of Mathematics. Boston, MA
  Pearson, 2004.
• http//scienceworld.wolfram.com
• http//www.groups.dcs.stand.ac.uk/history/mathema
  ticians/Euclid
• www.pde.state.pa.us