1. 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.

2. The Greek World Centered on the Aegean Sea

3. Greek History Outline ~1600 – 1100 BCE: Bronze Age Greece, Mycenaean civilization, last phase may have been time of the Iliad – to extent that it records actual history

4. Greek History Outline, Continued A very unsettled period after that – collapse of the New Kingdom in Egypt, Bronze Age Greece, and general unrest around Mediterranean. ~1100 – 750 B.C.E. The Greek “Dark Ages” (written language lost), oral traditions (including the epics) maintained 750 – 500 B.C.E. Archaic period (first half of 6 th 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) – will discuss briefly next time

5. 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

6. 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.

7. 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

8. 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.

9. 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”)

10. 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

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

12. 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 becomes foremost center of mathematical work in the world at this time. Famous Library and Museum or “university” were the focus.

13. The major focus in our study of Greek mathematics will be on Euclid Not much is known personally – no firm dates of birth or death, place of birth, etc. Proclus (~450 CE): “This man lived in the time of the first Ptolemy; for Archimedes, who followed closely on the first Ptolemy makes mention of Euclid …. He is therefore younger than Plato's circle but older than Eratosthenes and Archimedes …. In his aim he was a Platonist, …, whence he made the end of the whole Elements the construction of the so- called Platonic figures.”

14. Traditions and anecdotes Euclid trained at the Academy in Athens and then moved to Alexandria, where he had many students. Developed his most famous work, The Elements, as summary of basic mathematics known to his time, drawing on works of Eudoxus, Theaetetus, other earlier mathematicians. Elements was used as a textbook, from the start. Anecdotes about Euclid as a teacher also preserved(!)

15. But was there an actual historical Euclid? Possible scenarios proposed by historians: There was a historical figure named Euclid who wrote the Elements and other works attributed to him as an individual author A historical Euclid was leader of a group working in Alexandria who contributed jointly to works that were distributed under his name, possibly after his death The works of Euclid were written by a group of mathematicians who used the name of the philosopher Euclid of Megara (about 400 BCE)

16. The Elements The earliest known complete manuscripts date from about 900 CE – about 1200 years after Euclid's death(!) (Other earlier fragments too.) For a long time, most editions derived from a version with commentary by a later Alexandrian mathematician named Theon from about 400 CE. In 1808, an earlier version was discovered in the Vatican Library in Rome, with not too many differences – text was remarkably stable!

17. Two pages of the Vatican Euclid

18. 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.

19. Transmission history, continued In the 8 th Century C.E. many of these texts preserved in the Eastern (Byzantine) Empire were translated into Arabic, They passed from Islamic world to Western Europe in the 12 th century C.E.

20. Transmission history, concluded First English translation, 1570 CE

21. Final Comments on Euclid 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” and put them “out of business.”

22. Post-Euclid Greek Mathematics Archimedes (287 – 212 B.C.E.) Active in Syracuse in Sicily. Greatest mathematician of the ancient world (work foreshadows developments of calculus 1800 years later) Apollonius of Perga (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(!)