Overview of Greek History and
Greek Mathematics
MONT 104Q – Mathematical Journeys,
September 2015
The Greek World – pre-Alexander
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Centered on the Aegean Sea, modern-day
Greece and Turkey
Greek History Outline
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~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
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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
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“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
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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”
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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
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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
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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
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Sparta dominant until about 371 BCE.
Rise of Macedonia under Phillip (father of
Alexander), 350 – 340 BCE.
Alexander
Alexander ``the Great''
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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
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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
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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
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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(!)