Euclid’s Plane Geometry
The Elements
Euclid 300’s 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.
Aristotle 384-322 BCE
Begin your scientific
work with definitions
and axioms.
The Elements
► Consisted
of 13 volumes of definitions, axioms,
theorems and proofs.
► Compilation
► The
of knowledge.
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.
► 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
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…)
5 Common notions (obvious)
1.
Things equal to the same thing are equal.
2.
If equals are added to equals, the results
are equal.
3.
If equals are subtracted to equals, the
results are equal.
4.
Things that coincide are equal.
5.
The whole is greater than the part.
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.
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
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.
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.
2:6:18
Euclidean Algorithm
(Book VII)
►
Process for finding the greatest common divisor.
►
Given a, b with a > b, subtract b from a repeatedly until
get remainder c.
►
►
►
Then subtract c from b repeatedly until get to m, then
subtract m from c……when 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
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.
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
Solve this last equation for Sn
(ar – a):a = (arn-a):Sn
ar  a ar  a

a
Sn
n
a(r  1)
Sn 
r 1
n
Ex. Find the sum of the first 5 terms when a =1 and r =2
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.”
High School Geometry
►
Plane Geometry courses today are basically the content
of Euclid’s Elements.
►
Two-column proof appeared in the 1900’s to make
proofs easier but led to rote memorization instead.
►
1970’s 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.
PSSA
► Standards:
what they should know
► Anchors: what they are tested on
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 Heiberg’s text. This version
is the one most widely used and the basis for modern
Geometry courses.
►
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.
1981.
New York,
Katz, V. The History of Mathematics. Boston, MA:
Pearson, 2004.
http://scienceworld.wolfram.com
http://www.groups.dcs.stand.ac.uk/~history/mathematicians/Euclid
www.pde.state.pa.us