Greek Mathematics Period
(600 B.C.-A.D. 500)
Much of what is
considered Greek
math may not have
come from Greece,
but was written in
the language
A. Greek Logic and Philosophy
Greek philosophers promote logical, rational
explanations of natural phenomena.
Schools of logic, science and mathematics are
established.
Mathematics is viewed as more than a tool to
solve practical problems; it is seen as a
means to understand divine laws.
Mathematicians achieve fame, are valued for
their work.
B. Euclidean Geometry
The first mathematical system based on
postulates, theorems and proofs appears in
Euclid's Elements.

Development of deductive geometry (Thales, Pythagoras)

Start of number theory, Discovery of irrational numbers,
Geometric solution of Quadratic equations (Pythagorean
school)

Systematization of deductive logic (Aristotle 340 B.C.)

Geometry of conic sections (Apollonius,225 B.C.)

Axiomatic development of geometry, Algebraic identities
(Euclid,300 B.C)

Germ of the Integral calculus (Archimedes, 225 B.C)
First of the Greek mathematicians is Thales, the
traditional father of Greek math about 600BC, and
Pythagoras about 500BC.
Greek was very clumsy in writing down the numbers.
They didn’t like algebra.
They found it very hard to write down equations or
number problems.
Instead, Greek mathematicians were focused on
geometry, and used geometric methods to solve problems
that you might use algebra for.
Thales
Thales was born in the
city of Miletus, about
630 BC.
Thales travelled all over
when he was a young
man, and he may have
studied with Egyptian or
Babylonian scientists.
He figured out a way to
measure the heights of
the Egyptian pyramids
Thales may have been Anaximander's
teacher, and Anaximander was Pythagoras'
teacher.
Some ancient writers say that Pythagoras,
when he was young, actually visited Thales,
and that Thales advised Pythagoras to go
study in Egypt.
Thales died in 543 BC, only a few years after
his city was conquered by the Persians.
Thales proved that
• a circle is bisected by its diameter,
• the angles at the bases of any isosceles triangle are
equal
• if two straight lines cut one another, the opposite angles
are equal
• if two triangles have two angles and a side in common,
the triangles are identical
• An angle in a semicircle is a right angle.
• The sides of similar triangles are proportional.
• Two triangles are congruent if they have two angles and
a side respectively equal.
SEE: VIDEO 1 (Thales)
Pythagoras lived in the 500s
BC, and was one of the first
Greek mathematical thinkers.
He spent most of his life in the
Greek colonies in Sicily and
southern Italy. He had a group
of followers (like the later
disciples of Jesus) who followed
him around and taught other
people what he had taught them.
Pythagoreans were interested in philosophy,
but especially in music and mathematics, two
ways of making order out of chaos.
Music is noise that makes sense, and
mathematics is rules for how the world works.
The Sumerians, two thousand years earlier,
already knew that Pythagoras theorem was
generally true, and they used it in their
measurements, but Pythagoras is said to
have proved that it would always be true.
We don't really know whether it was Pythagoras
that proved it, because there's no evidence for it
until the time of Euclid, but that's the tradition.
Some people think that the proof must have been
written around the time of Euclid.
The Pythagoreans were known for their pure lives
(they didn't eat beans, for example, because they
thought beans were not pure enough). They wore
their hair long, and wore only simple clothing, and
went barefoot.
Pythagorean Triples
VIDEO 2
Euclid (c 300 BC), Alexandria
Euclid of Alexandria lived in 365 - 300 BC
(approximately).
Very little is known about Euclid's life except that he
taught in Alexandria, Egypt. He may have become
educated at Plato's Academy in Athens, or possibly from
some of Plato's students.
Basically, all of the rules we use in Geometry today are
based on the writings of Euclid, specifically 'The
Elements'. The Elements includes the following
Volumes:
Vol 1-6 : Plane Geometry
Vol 7-9 : Number Theory
Vol 10 : Eudoxus' Theory of Irrational Numbers
Vol 11-13 : Solid Geometry
Euclid's book the Elements also contains the beginnings
of number theory.
The Euclidean algorithm which is often referred to as
Euclid's algorithm is used to determine the greatest
common divisor (gcd) of two integers.
It is one of the oldest algorithms known, and was
included in Euclid's Elements.
Euclid's algorithm does not require factoring.
Euclid was older than Archimedes (287 - 212 B.C.E.)
and Eratosthenes and younger than Plato.
It is uncertain, but Euclid was probably a Greek who
had traveled to the city of Alexandria to learn and
teach.
Euclid and Demetrius Phalereus were invited to open
the mathematical school and to take charge of the
library, at the Museum and Library at Alexandria.
He probably wrote the Elements around 320 B.C.
thus leaving him around the age of 40.
No actual original copies exist from his time; the
earliest copy in existence is dated from 888 A.D.
You can find an excellent online edition of Euclid’s Elements in
the following web pages :
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
http://archimedes.fas.harvard.edu/euclid/digilib.html
Euclid's Elements form one of the most beautiful and
influential works of science in the history of
humankind.
Its beauty lies in its logical development of geometry
and other branches of mathematics.
It has influenced all branches of science but none so
much as mathematics and the exact sciences. The
Elements have been studied 24 centuries in many
languages starting, of course, in the original Greek,
then in Arabic, Latin, and many modern languages.
VIDEO 3
Euclid- Elements
ARCHIMEDES
Born about 287
BC in Syracuse,
Sicily.
At the time
Syracuse was
an independent
Greek city-state
with a 500-year
history.
Archimedes, died 212 or 211 BC in Syracuse when
it was being sacked by a Roman army.
He was killed by a Roman soldier who did not know
who he was.
He probably studied in Alexandria, Egypt, under the
followers of Euclid.
Generally regarded as the greatest mathematician
and scientist of antiquity and one of the three
greatest mathematicians of all time (together with
Isaac Newton (English 1643-1727) and Carl Friedrich
Gauss (German 1777-1855)).
He invented many war machines used in the defense
of Syracuse, compound pulley systems, planetarium,
water screw , water organ (possibly), burning mirrors
(very unlikely).
He is known as the father of Mathematical Physics.
See: VIDEO 4 (Samples of inventions)
See: VIDEO 5 (Water Screw)