Greek mathematics
Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant 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 Greek culture and the Greek language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēma Attic Greek: [má.tʰɛː.ma] Koine Greek: [ˈma.θi.ma], meaning "subject of instruction".^{ [1]} The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is an important difference between Greek mathematics and those of preceding civilizations.^{ [2]}^{ [3]}^{ [4]}
Origins of Greek mathematics
The origin of Greek mathematics is not well documented.^{ [5]}^{ [6]} 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 fourstory 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.^{ [7]}^{ [8]}^{ [5]} Unlike the flourishing of Greek literature in the span of 800 to 600 BC, not much is known about Greek mathematics in this early period—nearly all of the information was passed down through later authors, beginning in the mid4th century BC.^{ [9]}^{ [10]}
Archaic and Classical periods
Greek mathematics allegedly began with Thales of Miletus (c. 624–548 BC). Very little is known about his life and works, although it is generally agreed that he was one of the Seven Wise Men of Greece. According to Proclus, he traveled to Babylon from where he learned mathematics and other subjects, and came up with the proof of what is now called Thales' Theorem.^{ [11]}^{ [12]}
An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BC), who supposedly visited Egypt and Babylon,^{ [10]}^{ [13]} and ultimately settled in Croton, Magna Graecia, where he started a kind of cult. Pythagoreans believed that "all is number" and were keen in looking for mathematical relations between numbers and things.^{ [14]} Pythagoras himself was given credit for many later discoveries, including the construction of the five regular solids. However, Aristotle refused to attribute anything specifically to Pythagoras and only discussed the work of the Pythagoreans as a group.^{ [15]}^{ [16]}
It has been customary to credit almost half of the material in Euclid's Elements to the Pythagoreans, as well as the discovery of irrationals, attributed to Hippassus (c. 530450 BC), and the earliest attempt to square the circle, in the work of Hippocrates of Chios (c. 470410 BC).^{ [17]} The greatest mathematician associated with the group, however, may have been Archytas (c. 410350 BC), who solved the problem of doubling the cube, identified the harmonic mean, and possibly contributed to optics and mechanics.^{ [17]}^{ [18]} Other mathematicians active in this period, without being associated with any school, include Theodorus (fl. 450 BC), Theaetetus (c. 417369 BC), and Eudoxus (c. 408355 BC).
Greek mathematics also draw the attention of philosophers during the Classical period. Plato (c. 428348 BC), the founder of the Platonic Academy, mentions mathematics in several of his dialogues. While not considered a mathematician, Plato seems to have been influenced by Pythagorean ideas about number and believed that the elements of matter could be broken down into geometric solids.^{ [19]} He also believed that geometrical proportions bound the cosmos together rather than physical or mechanical forces.^{ [20]} Aristotle (c. 384322 BC), the founder of the Peripatetic school, often used mathematics to illustrate many of his theories, as when he used geometry in his theory of the rainbow and the theory of proportions in his analysis of motion.^{ [20]} Much of the knowledge known about ancient Greek mathematics in this period is thanks to records referenced by Aristotle in his own works.^{ [10]}^{ [21]}
Hellenistic and Roman periods
The Hellenistic era 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 the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to a Hellenistic mathematics.^{ [23]}^{ [24]}
Greek mathematics and astronomy reached its acme during the Hellenistic and early Roman periods, and much of the work represented by scholars such as Euclid (fl. 300 BC), Archimedes (c. 287212 BC), Apollonius (c. 240190 BC), Hipparchus (c. 190120 BC), and Ptolemy (c. 100170 AD) was of a very advanced level.^{ [25]} There is also evidence of combining mathematical knowledge with high levels of technical expertise, as found for instance in the works of Hero (c. 1070 AD) or in the construction of simple analogue computers like the Antikythera mechanism.^{ [26]}^{ [27]}
Several Hellenistic centers of learning appeared during this period, of which the most important one was the Musaeum in Alexandria, Egypt, which attracted scholars from across the Hellenistic world (mostly Greek, but also Egyptian, Jewish, Persian, Phoenician, and even Indian scholars).^{ [28]}^{ [29]} Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues.^{ [30]}
Later mathematicians include Diophantus (c. 214298 AD), who wrote on polygonal numbers and a work in premodern algebra ( Arithmetica),^{ [31]}^{ [32]} Pappus of Alexandria (c. 290350 AD), who compiled many important results in the Collection,^{ [33]} and Theon of Alexandria (c. 335405 AD) and his daughter Hypatia (c. 370415 AD), who edited Ptolemy's Almagest and other works.^{ [34]}^{ [35]} Although none of these mathematicians, save Diophantus, had notable original works, they are distinguished for their commentaries and expositions. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in the absence of original documents, are precious because of their rarity.^{ [36]}^{ [37]}
Most of the mathematical texts written in Greek survived through the copying of manuscripts over the centuries, though some fragments dating from antiquity have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.^{ [25]}
Achievements
Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.^{ [38]} Greek mathematicians also contributed to number theory, mathematical astronomy, combinatorics, mathematical physics, and, at times, approached ideas close to the integral calculus.
Eudoxus of Cnidus developed a theory of proportion that bears resemblance to the modern theory of real numbers using the Dedekind cut, developed by Richard Dedekind, who acknowledged Eudoxus as inspiration.^{ [39]}^{ [40]}^{ [41]}^{ [42]}
Euclid collected many previous results and theorems in the Elements, a canon of geometry and elementary number theory for many centuries.^{ [43]}^{ [44]}^{ [45]}
Archimedes was able to use the concept of the infinitely small in a way that anticipated modern ideas of the integral calculus.^{ [46]}^{ [47]} Using a technique dependent on a form of proof by contradiction, he could reach answers to problems with an arbitrary degree of accuracy, while specifying the limits within which the answers lay. This technique is known as the method of exhaustion, and he employed in several of his works, such as to approximate the value of π ( Measurement of the Circle).^{ [48]} 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. One of his two proofs shows the solution to the problem as an infinite geometric series, whose sum was 4/3.^{ [49]} In The Sand Reckoner, Archimedes set out to name 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.^{ [50]}
The most characteristic product of Greek mathematics may be the theory of conic sections, which was largely developed in the Hellenistic period, primarily by Apollonius.^{ [51]}^{ [52]}^{ [53]} The methods used made no explicit use of algebra, nor trigonometry, the latter appearing around the time of Hipparchus.^{ [54]}^{ [55]}
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.^{ [56]} The two major sources are
 Byzantine codices, written some 500 to 1500 years after their originals, and
 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.
Reviel Netz has counted 144 ancient exact scientific authors, of these only 29 are extant in Greek: Aristarchus, Autolycus, Philo of Byzantium, Biton, Apollonius, Archimedes, Euclid, Theodosius, Hypsicles, Athenaeus, Geminus, Hero, Apollodorus, Theon of Smyrna, Cleomedes, Nicomachus, Ptolemy, Gaudentius, Anatolius, Aristides Quintilian, Porphyry, Diophantus, Alypius, Damianus, Pappus, Serenus, Theon of Alexandria, Anthemius, Eutocius.^{ [57]}
Some works are extant only in Arabic translations:^{ [58]}^{ [59]}
 Apollonius, Conics books V to VII
 Apollonius, De Rationis Sectione
 Archimedes, Book of Lemmas
 Archimedes, Construction of the Regular Heptagon
 Diocles, On Burning Mirrors
 Diophantus, Arithmetica books IV to VII
 Euclid, On Divisions of Figures
 Euclid, On Weights
 Hero, Catoptrica
 Hero, Mechanica
 Menelaus, Sphaerica
 Pappus, Commentary on Euclid's Elements book X
 Ptolemy, Optics
 Ptolemy, Planisphaerium
See also
 Greek numerals
 Chronology of ancient Greek mathematicians
 History of mathematics
 Timeline of ancient Greek mathematicians
Notes
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References
 Boyer, Carl B. (1985), A History of Mathematics, Princeton University Press, ISBN 9780691023915
 Boyer, Carl B.; Merzbach, Uta C. (2011), A History of Mathematics (3rd ed.), John Wiley & Sons, Inc., ISBN 9780471543978
 Jean Christianidis, ed. (2004), Classics in the History of Greek Mathematics, Kluwer Academic Publishers, ISBN 9781402000812
 Cooke, Roger (1997), The History of Mathematics: A Brief Course, WileyInterscience, ISBN 9780471180821
 Derbyshire, John (2006), Unknown Quantity: A Real And Imaginary History of Algebra, Joseph Henry Press, ISBN 9780309096577
 Stillwell, John (2004), Mathematics and its History (2nd ed.), Springer Science + Business Media Inc., ISBN 9780387953366
 Burton, David M. (1997), The History of Mathematics: An Introduction (3rd ed.), The McGrawHill Companies, Inc., ISBN 9780070094659
 Heath, Thomas Little (1981) [First published 1921], A History of Greek Mathematics, Dover publications, ISBN 9780486240732
 Heath, Thomas Little (2003) [First published 1931], A Manual of Greek Mathematics, Dover publications, ISBN 9780486432311
 Szabo, Arpad (1978) [First published 1978], The Beginnings of Greek Mathematics, Reidel & Akademiai Kiado, ISBN 9789630514163
External links
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