Greek Mathematics Period ( 6 00 B.C.-A.D. 500)

1. 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

2. 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.

3. 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)

4. First of the Greek mathematicians isThales, the traditional father of Greek mathabout 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.

5. 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

6. 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.

7. 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 anglesand • a side respectively equal. • SEE: VIDEO 1 (Thales)

9. Pythagoras lived in the 500s BC, and was one of the first Greek mathematicalthinkers. 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) whofollowed him around and taught other people what he had taught them.

10. 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.

11. 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.

12. Pythagorean Triples VIDEO 2

13. Euclid (c 300 BC), Alexandria

14. 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 followingVolumes: Vol 1-6: Plane Geometry Vol 7-9: Number Theory Vol 10: Eudoxus' Theory of Irrational Numbers Vol 11-13: Solid Geometry

15. 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.

16. 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.

17. 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.

18. VIDEO 3 Euclid- Elements

19. ARCHIMEDES Born about 287 BC in Syracuse, Sicily. At the time Syracuse was an independent Greek city-state with a 500-year history.

20. 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)).

21. 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)