Overview of Greek History and Greek Mathematics MONT 104Q – Mathematical Journeys, September 2015

The Greek World – pre-Alexander Centered on the Aegean Sea, modern-day Greece and Turkey

Greek History Outline ~1600 to ~1200 BCE: Bronze Age Greece, Mycenaean civilization, last phase may have been time of the Iliad and Odyssey– to the extent that they record actual history

Greek History Outline, Continued A very unsettled period after that – collapse of the New Kingdom in Egypt, Bronze Age Greece, and general unrest around eastern Mediterranean ~1200 – 750 BCE. The Greek “Dark Ages” (written language lost), oral traditions (including the epics) maintained 750 – 500 BCE Archaic period (first half of 6th century BCE – Thales of Miletus; “birth of demonstrative mathematics,” Pythagoras born in Samos 572 BCE – moves to Crotona in Italy, founds Pythagorean brotherhood, dies after 500 BCE)

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

Greece and Persia 480 BCE. Another invasion attempt by Xerxes (son of Darius I), slowed up by Greeks at Thermopylae (depicted in “300”), Persians defeated again at Battles of Salamis, Plataea Our view of the Persians is colored by the Greeks' point of view (for instance by the Historia of Herodotus) – the victors write the histories(!) Greco-Persian wars continue until 449 BCE

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

Greek History, continued Ascendancy of Athens challenged by Sparta and other city states – Peloponnesian War 431 – 404 BCE – leads to defeat of Athens. Plato, ~425 – ~348 BCE: Academy founded in Athens 387 BCE (“Let no one unversed in geometry enter here”)

Mathematical Athens Plato's epistemology (philosophy of knowledge) put mathematics in a central role Athens also a “hotbed” of what we would call mathematical research: Eudoxus, 408-355 BCE – theory of proportions; developed “method of exhaustion,” a precursor of integral calculus Menaechmus, 380-320 BCE – work anticipating conic sections Aristotle, 384-322 BCE – not a mathematician as such but active in development of logic.

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

Alexander ``the Great'' Tutored by Aristotle (no record that he did any mathematics, though!) Seeking revenge, he finally crushes the Persian empire, conquers almost everything between the Mediterranean and India (336 BCE – 323 BCE). Dies in Mesopotamian city of Babylon. Founds the city of Alexandria in Egypt, 332 BCE.

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.

Euclid Not much known about him 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.”

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(!)

But was there a 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)

The Elements Earliest known complete manuscripts ~900 CE -- about 1200 years after Euclid's death. (Other earlier fragments too.) Most editions derive from a version with commentary by Theon, a later Alexandrian mathematician from about 400 CE -- 700 years after Euclid's time(!). In 1808, an earlier version was recognized in the Vatican Library in Rome, with not too many differences -- text was remarkably stable!

Two pages of the Vatican Euclid

A tangled transmission history Euclid's Elements were written in Greek, of course. The first Latin translations were made not from Greek sources, though, but from the Arabic. Reason: Euclid, and most other classical literature, was lost in Western Europe after the fall of the Western Roman Empire in 476 CE, only preserved in Byzantine (Eastern Roman) Empire and then transmitted through contacts with Islamic caliphate in Baghdad (8th century CE), Muslim Spain in the (12th century CE).

Transmission history, concluded First English translation, 1570 CE

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.” Study of Euclid was a traditional cornerstone of Jesuit education – Christopher Clavius, S.J. made a widely-used translation (published in 1627 CE after his death).

Post-Euclid Greek Mathematics Archimedes (287 – 212 B.C.E.) Active in Syracuse in Sicily. Greatest mathematician of the ancient world (building on Eudoxus, work foreshadows 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(!)