Overview of Greek History and Greek Mathematics MONT

Overview of Greek History and Greek Mathematics MONT 104 Q – 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 6 th 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 socalled 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(!)

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

The Elements, Book I – Axioms (Common Notions) and Postulates

The 5 Common Notions 1. Things that are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things that coincide with one another are equal to one another. 5. The whole is greater than the part.

The First Four Postulates 1. (It is possible) to draw a straight line from any point to any point. 2. (It is possible) to produce any finite straight line continuously in a straight line. 3. (It is possible) to describe a circle with any center and distance. 4. All right angles are congruent to one another.

The Fifth Postulate 5. If a straight line falling on two straight lines makes the angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The situation in Postulate 5

After Producing the lines sufficiently far, …

Comments 1. 2. 3. Postulates 1, 2, and 3 describe the constructions possible with an (unmarked) straightedge and a “collapsing” compass – that is the compass can be used to draw circles but not to measure or transfer distances Postulate 4 is a statement about the homogeneous nature of the plane – every right angle at one point is congruent to a right angle at any other point Postulate 5 is both more complicated than, and less “obvious” than the others(!)