Greek History before Euclid

Chapter 1: Before Thales

Really early: the Myceneans

The earliest Greek civilization was the Mycenean civilization.
It was roughly contemporary with the New Kingdom in Egypt
or the first half of the Assyrian age in Mesopotamia.
See timeline.

No known mathematics.
They had writing called Linear B.
Deciphered in 1952 by Michael Ventris (from England).
Not at all like the Greek alphabet,
but the language is clearly an early form of Greek.
Before Linear B was deciphered, no one knew for sure
if the Myceneans were Greek or not.

Mycenean artifacts:
The Mask of Agamemnon
Mycenean Throne room
Mycenean Art: Gold cup depicting a wild bull capture

The Trojan War

Took place around 1200 BC.
It was near the end of the Mycenaen period of Greek history.
The Trojans might have been decendents of the Hittites. (see timeline).
Lots of legends: for example, the Trojan Horse
Until Troy was discovered and excavated, some people
thought Troy was just a legend.

The Mycenean culture ended shortly after the Trojan war. (Eves: 1200-1150 BC).
About this time there was wholesale destruction of Mycenean sites.
Several hundred years of dark ages.

Homer

Homer at UVA

Tales of the Trojan war were preserved through an oral tradition.
The most famous Greek poet was Homer, who lived hundred of years after the Trojan war.
The Illiad is about the confrontation of Achilles and Hector in the Trojan war.
(Ilium is another name for Troy).
Homer wrote another epic: the Odyssey.
It concerns the journey home of Odysseus.
Here is a vase painting of Odysseus and the Sirens, a famous episode in the Odyssey.
Homer's epics were among the first written Greek texts.

Archaic Greek culture

Greeks dated their history from 776 BC: date of the first Olympic games.
This is a convenient starting date for "Archaic Greek'' culture.
The Archaic Greek culture dates from about 776 BC to the Persian-Greek wars.
It coincides more or less with the beginnings of writing and early coinage.
Trivia: the only event in the first Olympiad was the footrace.
(later wresting, discus, and other events were added. Even poetry and drama.)
The victorious Olympians became famous throughout Greece.

For us 776 BC is a convenient starting date for Archaic Greek culture
because it is about the time of
1. Reliable Greek history
2. Greek writing
3. Coinage
4. (Archaic) Greek art

The artistry of pottery reached a high level.
Here is Achilles and Ajax relaxing between battles.
Here is the famous (and rich) Lydian king Croesus suffering his fate after the Persians conquered his kingdom (legend has it that the Persian king Cyrus spared his life).
Here is the god Dionysus.

The Greek alphabet, modified from the Phoenician, was starting to become widely used in Greece.
The Phoenicians were Traders in the Mediterranean.
Here is a a captured Caananite or Phoenician.

The Greek alphabet is the first true alphabet since it has vowels.
The earlier Semitic writing systems, including that of the Phoenician, only had consonants.
(Here is the Greek alphabet after it was standardized.
Originally the Greek alphabet had local variations, and in many places it had letters not shown here.
Our alphabet comes from the Latin alphabet, which comes from an early western version of the Greek alphabet.)

Like the Phoenicians, the Greek also became maritime traders.
This famous plate of Dionysus illustrates both the maritime tradition (and hazards) as well as the religion of the time.

City States

At this time Greece was divided into city states.
The most famous are Sparta and Athens.
Miletus will be important for us as well.
They were often at war.
So there was a warrior culture. Soldiers were called hoplites.
(pronounced: "HOP light", but in Greek "hop LEE tace")
Sparta was known for its militarism.
Athens was eventually known for its culture and democracy.
Miletus was known for its wealth, and scientific tradition started by Thales.

Colonies and the Sea

Map

Importance of the sea and of sea-based trade.
Phoenician influence.
Greek farmland.
Olives and Grapes (Oil and Wine).

The Greeks were rapidly expanding along coasts of the Mediterranean. Building cities and temples all over the area. There contacts with other peoples greatly influenced the rapid development of their civilization.

The Greeks eventually developed a fierce navy (triremes).

Greek colonies were always near water.

Chapter 2: Early Greek Mathematics: Thales and Pythogoras

6th century BC: timeline.
The time of Thales and Pythagoras.
(No actual mathematical text survives from this period. We rely on later sources.)

Asia Minor

Map.

Croesus a king of Lydia, was not a Greek.
Much interaction between Greeks and their neighbors: exchange of ideas.
Prosperity: Early coinage
Ionia was a Greek area on the coast of Asia Minor.

Thales

Thales was an Ionia from Miletus.
Founder of Greek geometry and natural philosophy.
His most impressive result was that any triangle inscribed in a circle whose base is a diameter is a RIGHT TRIANGLE. This is called "Thales Theorem".

Thales is also given credit for four other theorem of geometry.
Said to have predicted the eclipse of 585 BC.
Credited with the first use of proof in mathematics.
Said to have brought geometry to Greece from Egypt.

Pythagoras

Pythagoras was the other major mathematical person from the period (he is pictured as wearing a turban, which was unusual for Greeks; perhaps this indicates his eastern travels).

Map: Samos, Kroton.

Pythagoras from the island of Samos.
Also lived in the 6th century, but later than Thales.
He is said to have travelled to Egypt and Babylon, and learned from priests.
After coming back to the Greek world he founded a mystical or religious society in Kroton.
He and his followers, called the Pythagoreans. were fascinated with mathematics and musical harmonies. Pythagoras said "all is number".
Discovered the simple mathematical ratios associated to harmonious notes in music.
He is given credit for the first proof of the Pythagorean theorem (the theorem, but perhaps not the proof, was known to the Babylonians).

Pythagorean Theorem

The statement (square of side plus square of side equals square of diagonal) was known to the Babylonians. The tablet Plimpton 322 gives an interesting list of Pythagorean triples. Here is a conversion into modern decimal notation.

Who came up with the first proof of this theorem? Perhaps Pythagoras did (and perhaps he sacrificed an Ox for his proof). Even if Pythagoras did come up with a proof, it was probably not as rigorous as later proofs.

Books I and II of Euclid (of the XIII total) are supposed to have been based on the Pythagoreans. Book I essentially closes with a proof of the Pythagorean theorem.

Figurative Numbers

Relates to the saying "all is number". Several nice identities can be illustrated using figurative numbers.

Pythagorean Triples

The Bablyonians probably knew the general formula, but the Pythagoreans came up with a formula (but not the general formula). Later Greeks had the general formula.

Golden Section

Example: diagonals of the pentagram.

Incommensurable Magnitudes

We say that two lengths x and y (or areas, or angles, or volumes) are "incommensurable" if there is not a common unit u such that x and y are both integer multiples of the unit u. (In modern terms, this is equivalent to x/y being irrational, where x and y can be measured using any unit whatsoever).
Example: diagonal of the square compared to a side.
Example: diagonal of the pentagon compared to a side.

Perfect Numbers

Odd perfect numbers. Do they exist? Are there an infinite or a finite number?

Musical Harmonies

Chapter 3: The Fifth Century

Timeline

The Persian Empire

Map of Persia

The Persian Empire was the larger than any previous empire.
It arose during the time of Thales and Pythagoras.
One of its capital's was Persepolis. Persian art was at a high level.
The most famous kings of the Empire were Cyrus, Darius, and Xerxes.
(The Greek historian Herodotus describes the Persian, and the Greek-Persian war in great detail. It makes extremely entertaining reading, even today!)

Expansion of the Persian Empire: key and map. Eventually Babylon, Egypt, and and Ionia were in the Empire. Subjects nations (as illustrated in the bas relief in Persepolis) had to pay large tributes to the Empire. This led to the Ionian revolt (see Herodotus for more details).

The Persian-Greek War

Soldiers

The Persian-Greek war occurred about 480 B.C. In this war the Greeks were able to hold off the Persians and maintain their independence. However, the Persian empire was still vast and powerful, but the Greeks were becoming more powerful and sophisticated as well. Famous battles include Thermopolai and Marathon. Athens, Sparta, and many other Greek city states united in this effort.

Age of Pericles and Socrates

The Greek victory initiated a golden age, especially in Athens.

Timeline.

Bust of Pericles.

The Parthenon.

After the war, Athens became the cultural leader of all of Greece. Philosophers such as Anaxagoras, and sophists such as Hippias flocked to Athens. This was the age of Pericles, Socrates, and Sophocles. Great literature, plays, philosophy, art, architecture, and mathematics were created.

Another view of the Parthenon.

The end of the century, however, was not as kind to Athens. The Peloponnesian war was long and costly, and eventually resulted in the defeat of Athens. Athens also suffered a terrible plague. Finally, in 399 B.C. Socrates was sentenced to death for "impiety".

Mathematicians

Timeline

The fifth and fourth centuries formed the "flowering of Greek mathematics" which culminated in Euclid's elements. Most of the results mentioned by Euclid were developed during this period.

The Pythagoreans were very active during this time.

Hippocrates of Chios was active in Athens about this time (he was not the famous physician Hippocrates from Cos from about the same time and who came up with the ``Hippocratic oath''). He was the author of the first surviving fragment of Greek mathematics. In it, he showed that a certain lune is equal in area to a certain triangle. Thus he showed that some curved regions could be squared. He was apparently trying to square the circle. He also wrote a book called the Elements (now lost). He was also interested in duplicating the cube. He did not solve it completely, but he did reduce the problem to the problem of two mean proportionals: This idea was later used by Archytas and others to find a solution.

Also active at this time was Hippias the sophist who lived in Athens. He invented the quadratrix which can be used to to trisect angles (or multisect angles) or to square the circle.

Chapter 4: Fourth Century

Timeline. This century witnessed an increase in mathematical sophistication.

Plato's Academy

Socrates (Louvre).

Although Socrates never wrote anything, his student Plato wrote quite a lot (but not nearly as much as Aristotle) and is still probably the most widely read philosopher. Socrates is the main character in Plato's dialogs. Much of what Socrates says in the dialogs is historically Socrates own views, but much of what he says is Plato's view. There are several interesting mathematical discussions in Plato's dialogs.

Death of Socrates (Jacques-Louis David, 1787).
Socrates died in 399 BC at the start of the 4th century BC.

Plato and his friend Theaetetus of Athens travelled to North Africa to the Greek colony of Cyrene (also written Kyrene). There they studied under Theodorus, who might have been a later Pythagorean (at a time when Pythagoreans were apparently not sworn to secrecy).

Plato (Raphael 1509).

Plato also travelled to Italy and became friends with Archytas of Tarentum. Archytas was a later Pythagorean, and also a general and statesman. He duplicated the cube by the method of two mean proportions. His method for constructing two mean proportions is based on lengths which arise by intersecting a certain cone, cylinder, and torus in a certain way.

Academy (This picture was painted by Raphael in the Renaissance)

So Plato was a big fan of mathematics. Even though he didn't do much mathematics himself. Plato founded the Academy and of course Mathematics was a central, and lively topic of study there.

Plato's friend Theaetetus of Athens was one of the most important mathematicians of Plato's Academy. He was important in the study of incommensurables, and the so called Platonic solids. Some say he discovered the octahedron and icosahedron. (We will discuss other mathematicians of Plato's academy, including Eudoxus, in the next unit).

One of Plato's students was Aristole, who later founded his own school called the Lyceum, and became a personal tutor to Alexander the great.

Chapter 5: Alexander the Great and Euclid

Timeline and Map (Macedonia).

Macedonia is a little north of Greece, and its young king Alexander was tutored by Aristotle. Alexander thought that he should conquer and spread Greek culture throughout the known world.

His Empire.

When Alexander died in 323 B.C., his empire was split into smaller kingdoms led by his former generals. The Seleucid dynasty, founded by Seleucus, held power in Mesopotamia (and Babylon was no longer the capital), and the Ptolemaic dynasty ruled Egypt, founded by Ptolemy. The capital of Egypt became Alexandria. There were other kingdoms as well.

Alexandria became the cultural capital of the Hellenistic world, displacing Athens and Babylon in importance (however, the Academy and Lyceum continued to exist). It was here where Euclid wrote his Elements in about 300 BC. Euclid organized the ideas of the Pythagoreans (both Geometry and Higher Arithmetic), Hippocrates (circles), Theaetetus (incommensurables), Eudoxus (proportions), and others into the most famous Geometry book of all times. The Elements culminated in the construction of the five "Platonic" solids. It does not cover all of Greek mathematics. For example, the detailed properties of the Conic sections would have to wait for Apollonius.

Index of Slides

Persian Slides

Cyrus Conquered Babylon
Darius
Map of Persia. Color coded map (with key).
Ruins of Persepolis.
rhyton.
Subjects of the Persian empire.

Greek Slides

Map of Greece and the Greek colonies.
Map Alexander. Another. Yet another.
Achilles Game
Croesus
Dionysus Boat
Euclid
Lune
Olympian
Parthenon (and another).
Pythagoras
Famous painting by Raphael: School of Athens (1510-11).
Thales
Alexander
Platonic Solids
Temple
timeline
trireme