Greek Mathematics – Overview We now turn to the mathematics of the next ancient civilization we will consider – the Greeks. Recall that both the height of ancient Egyptian mathematics (or at least the time of the surviving records) and the Old Babylonian period were roughly 2000 BCE to 1600 BCE. There was a very unsettled period about 400- 500 years after that – collapse of the New Kingdom in Egypt, Bronze Age Greece, and general unrest around Mediterranean.

Greek History Outline ~1600 – 1100 BCE: Bronze Age Greece, Mycenaean civilization, last phase was the presumed time of the Homeric epics (Iliad, Odyssey)

The Greek World Centered on the Aegean Sea

Greek History Outline, Continued ~1100 – 750 B.C.E. The Greek “Dark Ages” (written language lost), oral traditions maintained 750 – 500 B.C.E. Archaic period (first half of 6th century B.C.E. – Thales of Miletus; “birth of demonstrative mathematics,” Pythagoras born in Samos 572 B.C.E. – moves to Crotona in Italy, founds Pythagorean brotherhood, dies after 500 B.C.E)

Another very “eventful” history “Classical Period” – ~500 B.C.E. – 323 B.C.E. (death of Alexander the Great) Greece invaded by Persians under Darius I, 490 B.C.E. – Darius defeated at Battle of Marathon

Greece and Persia 480 B.C.E. Another invasion attempt by Xerxes (son of Darius I), slowed up by Greeks at Thermopylae (depicted in “300”) defeated again at Battles of Salamis, Plataea “300” had a totally inaccurate portrayal of the Persians, by the way. They had a very advanced and tolerant approach to ruling their subject peoples – very “civilized” Greco-Persian wars continue until 449 B.C.E.

Athenian “Golden Age” The fifty years or so between the defeat of the Persians under Xerxes and the start of the Peloponnesian War were the age of Pericles, Socrates in Athens. The Parthenon in Athens

Mathematical Athens Also a “hotbed” of mathematics (displaced Pythagoreans, Anaxagoras, Zeno, Parmenides, … ) Eudoxus – theory of proportions leading to our modern way of dealing with fractions Menaechmus – ``invented'' conic sections Aristotle – not a mathematician as such but very active in development of logic.

Greek History, continued Ascendancy of Athens challenged by Sparta and other city states – Peloponnesian War 431 – 404 B.C.E. – leads to defeat of Athens. Plato's Academy founded in Athens 387 B.C.E. (“Let no one unversed in geometry enter here”)

Greek History, Continued Sparta dominant until about 371 B.C.E. Rise of Macedonia under Phillip (father of Alexander) – 350 B.C.E. – 340 B.C.E. Alexander

Alexander Tutored by Aristotle (no record that he did any mathematics, though!) Alexander crushes the Persian empire, conquers Egypt, Mesopotamia, almost everything between the Mediterranean and India (336 B.C.E. – 323 B.C.E.) Dies in Mesopotamian city of Babylon. Alexander founds the city of Alexandria in Egypt, 332 B.C.E

Greek History, Continued After his death, Alexander's empire is divided between several of his generals, who found dynasties that last through the Hellenistic Period – Ptolemaic dynasty in Egypt, Seleucid dynasty in Syria and Mesopotamia Alexandria in Egypt becomes foremost center of mathematical work in the world at this time. Famous library and “university” were the focus.

Euclid Not much is known personally about Euclid – dates, place of birth, etc. He was apparently trained at the Academy in Athens and asked to head the “mathematics department” in Alexandria around 300 B.C.E. Developed the Elements apparently as a summary of basic mathematics known to the Greeks of his time and as a textbook. No text of the Elements survives from Euclid's own time.

The Elements of Euclid In fact the earliest known manuscripts date from about 700 years after Euclid's time(!) Modern editions are mostly derived from a version with commentary by a later Alexandrian mathematician named Theon from about 400 C.E. But in 1808, an earlier version was discovered in the Vatican Library in Rome, with not too many differences – text was remarkably stable!

A tangled transmission history The earliest versions of the Elements were written in Greek, of course. The first Latin translations were made not from Greek sources, though, but from Arabic sources. Euclid and most other classical literature was lost in Western Europe after the fall of the Western Roman Empire in 476 C.E.

Transmission history, continued In the 8th Century C.E. many of these texts preserved in the Eastern (Byzantine) Empire were translated into Arabic, and passed from there to Western Europe in the 12th century C.E. First Latin translation by Adelard of Bath (England) in 1120 C.E. First printed mathematics book – The Elements in Venice, 1482 C.E.

Transmission history, concluded First English translation, 1570 C.E. Some other comments: There were other Elements before Euclid's (Plato's Academy used a geometry text by a mathematician named Theudius, for instance.) None of them survive! Euclid quickly superseded all those predecessors and “competitors.”

Post-Euclid Greek Mathematics Archimedes (287 – 212 B.C.E.) Active in Syracuse in Sicily. Greatest mathematician of the ancient world (foreshadows developments of calculus 1800 years later) Apollonius (Alexandria: 262 – 190 B.C.E.) – deeper study of conic sections, other geometrical loci Diophantus (Alexandria: dates uncertain)– algebra and number theory Many others – almost all of them learned their basic mathematics from Euclid's Elements(!)