Greek Mathematics – Overview
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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.
The Greek World
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Centered on the Aegean Sea
Greek History Outline
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~1600 – 1100 BCE: Bronze Age Greece,
Mycenaean civilization, last phase may have
been time of the Iliad – to extent that it records
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
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
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) – will discuss briefly next time
Another very “eventful” history
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“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
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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”
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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
Mathematical Athens
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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
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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
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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
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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.
Greek 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.
The major focus in our study of
Greek mathematics will be on Euclid
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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 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(!)
But was there an actual historical
Euclid?
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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
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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!
Two pages of the Vatican Euclid
A tangled transmission history
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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
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In the 8th 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 12th century C.E.
Transmission history, concluded
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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.”
Post-Euclid Greek Mathematics
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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(!)