Greek mathematics, as that term is used in this article, is the
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
written in
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and
languageGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
. Greek mathematics of the period following
Alexander the Great is sometimes called Hellenistic mathematics. The word "mathematics" itself derives from the ancient Greek
μάθημα (
mathema), 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 origins of Greek mathematics are not easily documented. The earliest advanced civilizations in the country of
GreeceGreece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....
and in
EuropeEurope is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
were the
MinoanThe Minoan civilization was a Bronze Age civilization that arose on the island of Crete and flourished from approximately the 27th century BC to the 15th century BC. It was rediscovered at the beginning of the 20th century through the work of the British archaeologist Arthur Evans...
and later Mycenean civilization, 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
EgyptianAncient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
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
ThalesThales of Miletus was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...
of
MiletusMiletus was an ancient Greek city on the western coast of Anatolia , near the mouth of the Maeander River in ancient Caria...
(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 Theorem of Thales, 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
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
of
SamosSamoš is a village in Serbia. It is situated in the Kovačica municipality, in the South Banat District, Vojvodina province. The village has a Serb ethnic majority and its population numbering 1,247 people .-See also:...
(ca. 580 - 500 BC). Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of
NebuchadnezzarNebuchadnezzar was the name of several kings of Babylonia.* Nebuchadnezzar I, who ruled the Babylonian Empire in the 12th century BC* Nebuchadnezzar II , the Babylonian ruler mentioned in the biblical Book of Daniel...
, but settled in
CrotonCrotone is a city and comune in Calabria, southern Italy, on the Ionian Sea. Founded circa 710 BC as the Achaean colony of Croton , it was known as Cotrone from the Middle Ages until 1928, when its name was changed to the current one. In 1994 it became the capital of the newly established...
,
Magna GraeciaMagna Græcia is the name of the coastal areas of Southern Italy on the Tarentine Gulf that were extensively colonized by Greek settlers; particularly the Achaean colonies of Tarentum, Crotone, and Sybaris, but also, more loosely, the cities of Cumae and Neapolis to the north...
. 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
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
.
Distinguishing the work of Thales and Pythagoras from that of later and earlier mathematicians is difficult since none of their original works survives, 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 in fact 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
ProclusProclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
’ commentary on
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
written centuries later. Some of these later works, such as
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments.
Thales is supposed to have used
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
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 a geometric theorem - the "Theorem of Thales" described above. Pythagoras is widely credited with recognizing the mathematical basis of musical
harmonyIn music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
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 theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
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
numerologyNumerology is any study of the purported mystical relationship between a count or measurement and life. It has many systems and traditions and beliefs...
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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
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
The
Hellenistic periodThe Hellenistic period or Hellenistic era describes the time which followed the conquests of Alexander the Great. It was so named by the historian J. G. Droysen. During this time, Greek cultural influence and power was at its zenith in Europe and Asia...
began in the 4th century BC with
Alexander the Great's conquest of the eastern Mediterranean,
EgyptAncient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
,
MesopotamiaMesopotamia is a toponym for the area of the Tigris–Euphrates river system, largely corresponding to modern-day Iraq, northeastern Syria, southeastern Turkey and southwestern Iran.Widely considered to be the cradle of civilization, Bronze Age Mesopotamia included Sumer and the...
, the
Iranian plateauThe Iranian plateau, or Iranic plateau, is a geological formation in Southwest Asia. It is the part of the Eurasian Plate wedged between the Arabian and Indian plates, situated between the Zagros mountains to the west, the Caspian Sea and the Kopet Dag to the north, the Hormuz Strait and Persian...
,
Central AsiaCentral Asia is a core region of the Asian continent from the Caspian Sea in the west, China in the east, Afghanistan in the south, and Russia in the north...
, and parts of
IndiaIndia , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
, 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
EgyptianEgyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...
and
Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...
to give rise to a Hellenistic mathematics.
The most important centre of learning during this period was
AlexandriaAlexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
in
EgyptAncient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
, which attracted scholars from across the Hellenistic world, mostly Greek and
EgyptianEgyptians are nation an ethnic group made up of Mediterranean North Africans, the indigenous people of Egypt.Egyptian identity is closely tied to geography. The population of Egypt is concentrated in the lower Nile Valley, the small strip of cultivable land stretching from the First Cataract to...
, but also Jewish,
PersianThe Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
,
PhoeniciaPhoenicia , was an ancient civilization in Canaan which covered most of the western, coastal part of the Fertile Crescent. Several major Phoenician cities were built on the coastline of the Mediterranean. It was an enterprising maritime trading culture that spread across the Mediterranean from 1550...
n and even
IndianThe history of India begins with evidence of human activity of Homo sapiens as long as 75,000 years ago, or with earlier hominids including Homo erectus from about 500,000 years ago. The Indus Valley Civilization, which spread and flourished in the northwestern part of the Indian subcontinent from...
scholars.
Most of the mathematical texts written in Greek have been found in Greece,
EgyptEgypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
,
Asia MinorAnatolia is a geographic and historical term denoting the westernmost protrusion of Asia, comprising the majority of the Republic of Turkey...
,
MesopotamiaMesopotamia is a toponym for the area of the Tigris–Euphrates river system, largely corresponding to modern-day Iraq, northeastern Syria, southeastern Turkey and southwestern Iran.Widely considered to be the cradle of civilization, Bronze Age Mesopotamia included Sumer and the...
, and
SicilySicily is a region of Italy, and is the largest island in the Mediterranean Sea. Along with the surrounding minor islands, it constitutes an autonomous region of Italy, the Regione Autonoma Siciliana Sicily has a rich and unique culture, especially with regard to the arts, music, literature,...
.
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
was able to use
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s in a way that is similar to modern
integral calculusIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in 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 exhaustionThe method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...
, and he employed it to approximate the value of
π' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
(Pi). In
The Quadrature of the ParabolaThe Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment is 4/3 that of a certain inscribed...
, Archimedes proved that the area enclosed by a
parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
and a straight line is times the area of a
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
with equal base and height. He expressed the solution to the problem as an infinite
geometric series, whose sum was . In
The Sand ReckonerThe Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...
, 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
myriadMyriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...
, which denoted 10,000.
Greek mathematics and astronomy reached a rather advanced stage during
HellenismThe Hellenistic period or Hellenistic era describes the time which followed the conquests of Alexander the Great. It was so named by the historian J. G. Droysen. During this time, Greek cultural influence and power was at its zenith in Europe and Asia...
, represented by scholars such as
HipparchusHipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
, Appolonius and
PtolemyClaudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
, to the point of constructing simple analogue computers such as the
Antikythera mechanismThe Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...
.
Achievements
Greek mathematics constitutes a major period in the history of
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, fundamental in respect of
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and the idea of
formal proofA formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...
. Greek mathematics also contributed importantly to ideas on
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
,
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
,
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, and, at times, approached close to integral calculus.
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
,
fl.Floruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...
300 BC, collected the mathematical knowledge of his age in the
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, a canon of geometry and elementary number theory for many centuries.
The most characteristic product of Greek mathematics may be the theory of
conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s, largely developed in the Hellenistic period. The methods used made no explicit use of
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, nor
trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
.
Eudoxus of CnidusEudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...
developed a theory of real numbers strikingly similar to the modern theory developed by
DedekindIn mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....
, who indeed acknowledged Eudoxus as inspiration.
Transmission and the manuscript tradition
Although the earliest
Greek languageGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
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
Syriac is a dialect of Middle Aramaic that was once spoken across much of the Fertile Crescent. Having first appeared as a script in the 1st century AD after being spoken as an unwritten language for five centuries, Classical Syriac became a major literary language throughout the Middle East from...
or Arabic translationsThe Translation Movement was a movement started in the House of Wisdom in Baghdad which translated many Greek classics into Arabic.The relationship between the early period of Islamic mathematics and the mathematics of Greece and India is not yet fully understood as much work is extant only in...
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 Baylonian 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
External links