Greek mathematics refers to mathematics texts
and advances written in Greek, developed from
the 7th century BC to the 4th century AD around
the shores of the Eastern Mediterranean. Greek
mathematicians lived in cities spread over
the entire Eastern Mediterranean from Italy
to North Africa but were united by culture
and language. Greek mathematics of the period
following Alexander the Great is sometimes
called Hellenistic mathematics. The word "mathematics"
itself derives from the Ancient Greek: μάθημα,
translit. máthēma Attic Greek: [má.tʰɛː.ma]
Koine Greek: [ˈma.θi.ma], meaning "subject
of instruction". The study of mathematics
for its own sake and the use of generalized
mathematical theories and proofs is the key
difference between Greek mathematics and those
of preceding civilizations.
== Origins of Greek mathematics ==
The origin of Greek mathematics is not well
documented. The earliest advanced civilizations
in Greece and in Europe were the Minoan and
later Mycenaean civilizations, both of which
flourished during the 2nd millennium BC. While
these civilizations possessed writing and
were capable of advanced engineering, including
four-story palaces with drainage and beehive
tombs, they left behind no mathematical documents.
Though no direct evidence is available, it
is generally thought that the neighboring
Babylonian and Egyptian civilizations had
an influence on the younger Greek tradition.
Between 800 BC and 600 BC, Greek mathematics
generally lagged behind Greek literature,
and there is very little known about Greek
mathematics from this period—nearly all
of which was passed down through later authors,
beginning in the mid-4th century BC.
== Classical period ==
Historians traditionally place the beginning
of Greek mathematics proper to the age of
Thales of Miletus (ca. 624–548 BC). Little
is known about the life and work of Thales,
so little indeed that his date of birth and
death are estimated from the eclipse of 585
BC, which probably occurred while he was in
his prime. Despite this, it is generally agreed
that Thales is the first of the seven wise
men of Greece. The two earliest mathematical
theorems, Thales' theorem and Intercept theorem
are attributed to Thales. The former, which
states that an angle inscribed in a semicircle
is a right angle, may have been learned by
Thales while in Babylon but tradition attributes
to Thales a demonstration of the theorem.
It is for this reason that Thales is often
hailed as the father of the deductive organization
of mathematics and as the first true mathematician.
Thales is also thought to be the earliest
known man in history to whom specific mathematical
discoveries have been attributed. Although
it is not known whether or not Thales was
the one who introduced into mathematics the
logical structure that is so ubiquitous today,
it is known that within two hundred years
of Thales the Greeks had introduced logical
structure and the idea of proof into mathematics.
Another important figure in the development
of Greek mathematics is Pythagoras of Samos
(ca. 580–500 BC). Like Thales, Pythagoras
also traveled to Egypt and Babylon, then under
the rule of Nebuchadnezzar, but settled in
Croton, Magna Graecia. Pythagoras established
an order called the Pythagoreans, which held
knowledge and property in common and hence
all of the discoveries by individual Pythagoreans
were attributed to the order. And since in
antiquity it was customary to give all credit
to the master, Pythagoras himself was given
credit for the discoveries made by his order.
Aristotle for one refused to attribute anything
specifically to Pythagoras as an individual
and only discussed the work of the Pythagoreans
as a group. One of the most important characteristics
of the Pythagorean order was that it maintained
that the pursuit of philosophical and mathematical
studies was a moral basis for the conduct
of life. Indeed, the words philosophy (love
of wisdom) and mathematics (that which is
learned) are said to have been coined by Pythagoras.
From this love of knowledge came many achievements.
It has been customarily said that the Pythagoreans
discovered most of the material in the first
two books of Euclid's Elements.
Distinguishing the work of Thales and Pythagoras
from that of later and earlier mathematicians
is difficult since none of their original
works survive, except for possibly the surviving
"Thales-fragments", which are of disputed
reliability. However many historians, such
as Hans-Joachim Waschkies and Carl Boyer,
have argued that much of the mathematical
knowledge ascribed to Thales was developed
later, particularly the aspects that rely
on the concept of angles, while the use of
general statements may have appeared earlier,
such as those found on Greek legal texts inscribed
on slabs. The reason it is not clear exactly
what either Thales or Pythagoras actually
did is that almost no contemporary documentation
has survived. The only evidence comes from
traditions recorded in works such as Proclus’
commentary on Euclid written centuries later.
Some of these later works, such as Aristotle’s
commentary on the Pythagoreans, are themselves
only known from a few surviving fragments.
Thales is supposed to have used geometry to
solve problems such as calculating the height
of pyramids based on the length of shadows,
and the distance of ships from the shore.
He is also credited by tradition with having
made the first proof of two geometric theorems—the
"Theorem of Thales" and the "Intercept theorem"
described above. Pythagoras is widely credited
with recognizing the mathematical basis of
musical harmony and, according to Proclus'
commentary on Euclid, he discovered the theory
of proportionals and constructed regular solids.
Some modern historians have questioned whether
he really constructed all five regular solids,
suggesting instead that it is more reasonable
to assume that he constructed just three of
them. Some ancient sources attribute the discovery
of the Pythagorean theorem to Pythagoras,
whereas others claim it was a proof for the
theorem that he discovered. Modern historians
believe that the principle itself was known
to the Babylonians and likely imported from
them. The Pythagoreans regarded numerology
and geometry as fundamental to understanding
the nature of the universe and therefore central
to their philosophical and religious ideas.
They are credited with numerous mathematical
advances, such as the discovery of irrational
numbers. Historians credit them with a major
role in the development of Greek mathematics
(particularly number theory and geometry)
into a coherent logical system based on clear
definitions and proven theorems that was considered
to be a subject worthy of study in its own
right, without regard to the practical applications
that had been the primary concern of the Egyptians
and Babylonians.
== Hellenistic and Roman periods ==
The Hellenistic period began in the 4th century
BC with Alexander the Great's conquest of
the eastern Mediterranean, Egypt, Mesopotamia,
the Iranian plateau, Central Asia, and parts
of India, leading to the spread of the Greek
language and culture across these areas. Greek
became the language of scholarship throughout
the Hellenistic world, and Greek mathematics
merged with Egyptian and Babylonian mathematics
to give rise to a Hellenistic mathematics.
Greek mathematics and astronomy reached a
rather advanced stage during the Hellenistic
and Roman period, represented by scholars
such as Hipparchus, Apollonius and Ptolemy,
to the point of constructing simple analogue
computers such as the Antikythera mechanism.
The most important centre of learning during
this period was Alexandria in Egypt, which
attracted scholars from across the Hellenistic
world, mostly Greek and Egyptian, but also
Jewish, Persian, Phoenician and even Indian
scholars.Most of the mathematical texts written
in Greek have been found in Greece, Egypt,
Asia Minor, Mesopotamia, and Sicily.
Archimedes was able to use infinitesimals
in a way that is similar to modern integral
calculus. Using a technique dependent on a
form of proof by contradiction he could give
answers to problems to an arbitrary degree
of accuracy, while specifying the limits within
which the answer lay. This technique is known
as the method of exhaustion, and he employed
it to approximate the value of π (Pi). In
The Quadrature of the Parabola, Archimedes
proved that the area enclosed by a parabola
and a straight line is 4/3 times the area
of a triangle with equal base and height.
He expressed the solution to the problem as
an infinite geometric series, whose sum was
4/3. In The Sand Reckoner, Archimedes set
out to calculate the number of grains of sand
that the universe could contain. In doing
so, he challenged the notion that the number
of grains of sand was too large to be counted,
devising his own counting scheme based on
the myriad, which denoted 10,000.
== Achievements ==
Greek mathematics constitutes a major period
in the history of mathematics, fundamental
in respect of geometry and the idea of formal
proof. Greek mathematics also contributed
importantly to ideas on number theory, mathematical
analysis, applied mathematics, and, at times,
approached close to integral calculus.
Euclid, fl. 300 BC, collected the mathematical
knowledge of his age in the Elements, a canon
of geometry and elementary number theory for
many centuries.
The most characteristic product of Greek mathematics
may be the theory of conic sections, largely
developed in the Hellenistic period. The methods
used made no explicit use of algebra, nor
trigonometry.
Eudoxus of Cnidus developed a theory of real
numbers strikingly similar to the modern theory
of the Dedekind cut developed by Richard Dedekind,
who indeed acknowledged Eudoxus as inspiration.
== Transmission and the manuscript tradition
==
Although the earliest Greek language texts
on mathematics that have been found were written
after the Hellenistic period, many of these
are considered to be copies of works written
during and before the Hellenistic period.
The two major sources are
Byzantine codices written some 500 to 1500
years after their originals
Syriac or Arabic translations of Greek works
and Latin translations of the Arabic versions.Nevertheless,
despite the lack of original manuscripts,
the dates of Greek mathematics are more certain
than the dates of surviving Babylonian or
Egyptian sources because a large number of
overlapping chronologies exist. Even so, many
dates are uncertain; but the doubt is a matter
of decades rather than centuries.
== See also ==
Greek numerals
Chronology of ancient Greek mathematicians
History of mathematics
Timeline of ancient Greek mathematicians
== Notes
