The Hellenistic period began in the 4th century BC with Alexander's conquest of the Eastern Mediterranean, including Egypt, as well as Mesopotamia and the Iranian plateau. Greek became the language of scholarship throughout the Hellenistic world, and Greek mathematics merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics. In turn, Hellenistic mathematics had an influence on Islamic mathematics (which merged it with Indian mathematics). As such, Hellenistic mathematics played an important role in the history of mathematics.
Overview
and i think that the most important centre of learning during this period was Alexandria in Egypt, which attracted scholars from across the Hellenistic world, including Greek, Egyptian, Jewish, Persian, Phoenician and even Indian scholars.^{[1]}
Most of the mathematical texts written in Greek have been found in Egypt, as well as Greece, Asia Minor, Mesopotamia, and Sicily.
Greek mathematics and astronomy reached a rather advanced stage during Hellenism, represented by scholars such as Hipparchus from Asia Minor and Ptolemy from Egypt.
Hellenistic Egypt (c. 300 BC—AD 400)
The most important centre of Hellenistic learning during this period was Alexandria in Egypt, which attracted scholars from across the Hellenistic world, including Egyptian, Greek, Jewish, Persian, Phoenician and even Indian scholars.^{[1]}
Most of the mathematical texts written in Greek have been found in Egypt, as well as Greece, Asia Minor, Mesopotamia, and Sicily.
Demographics
Under GrecoRoman rule, Egypt hosted several Greek settlements, mostly concentrated in Alexandria, but also in a few other cities, where Greek settlers lived alongside some seven to ten million native Egyptians.^{[2]} Faiyum's earliest Greek inhabitants were soldierveterans and cleruchs (elite military officials) who were settled by the Ptolemaic kings on reclaimed lands.^{[3]}^{[4]} Native Egyptians also came to settle in Faiyum from all over the country, notably the Nile Delta, Upper Egypt, Oxyrhynchus and Memphis, to undertake the labor involved in the land reclamation process, as attested by personal names, local cults and recovered papyri.^{[5]} It is estimated that as much as 30 percent of the population of Faiyum was Greek during the Ptolemaic period, with the rest being native Egyptians.^{[6]} By the Roman period, much of the "Greek" population of Faiyum was madeup of either Hellenized Egyptians or people of mixed EgyptianGreek origins.^{[7]} By the time of Roman emperor Caracalla in the 2nd century CE, the only way to differentiate Alexandria's "Greeks" from "genuine" ethnic Egyptians was "by their speech."^{[8]}
While commonly believed to represent Greek settlers in Egypt,^{[9]}^{[10]} the Faiyum mummy portraits instead reflect the complex synthesis of the predominant Egyptian culture and that of the elite Greek minority in the city.^{[6]} According to Walker, the early Ptolemaic Greek colonists married local women and adopted Egyptian religious beliefs, and by Roman times, their descendants were viewed as Egyptians by the Roman rulers, despite their own selfperception of being Greek.^{[11]} The dental morphology^{[12]} of the Romanperiod Faiyum mummies was also compared with that of earlier Egyptian populations, and was found to be "much more closely akin" to that of ancient Egyptians than to Greeks or other European populations.^{[13]}
The mathematics historian Victor J. Katz has criticized the modern misconception portraying the ancient Hellenistic scholars of Egypt as ethnically Greek or European, writing:^{[14]}
But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privaleged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist.
Ptolemaic Egypt (c. 300—30 BC)
In Ptolemaic Egypt, the Hellenistic Egyptian mathematician Euclid (c. 300 BC) is an early example of the format still used in mathematics today, definition, axiom, theorem, proof. He also studied conics. His book, Elements, was known to all educated people in the West until the middle of the 20th century.^{[16]} In addition to the familiar theorems of geometry, such as the Pythagorean theorem, Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.
The Sieve of Eratosthenes (ca. 230 BC), by Eratosthenes (Libyan mathematician active in Egypt), was used to discover prime numbers.
Roman Egypt (c. 30 BC—AD 400)
In Roman Egypt, the Hellenized Egyptian/Phoenician geometer Heron described a method of iteratively computing the square root.^{[17]} It is called the Babylonian method, because the Babylonians knew of it before Heron wrote it down.
The Hellenized Egyptian mathematician Menelaus of Alexandria (c. 100 AD) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.^{[18]} He establishes a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.^{[18]} Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.^{[18]} Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".^{[18]} He further gave his famous "rule of six quantities".^{[19]}
The Hellenized Egyptian astronomer Claudius Ptolemy (ca. 90  ca. 168 A.D.) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity.^{[20]} A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sumanddifference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.^{[20]} Ptolemy further derived the equivalent of the halfangle formula $ \sin^2({x/2}) = \frac{1  \cos(x)}{2} $.^{[20]} Ptolemy used these results to create trigonometric tables.^{[20]}
Diophantus was a Hellenized Babylonian/Egyptian/Jewish mathematician who lived in Alexandria, Egypt, circa 250 AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.^{[21]} Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, but resembles Babylonian mathematics to a much greater extent.^{[22]} In Arithmetica, Diophantus used symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;^{[22]} thus he used what is now known as syncopated algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.^{[23]}
Hellenistic Sicily (c. 300—200 BC)
Some say the greatest of Greek mathematicians, if not of all time, was Archimedes (c. 287—212 BC) of Syracuse. He used a method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave fairly accurate approximations of Pi.^{[24]} He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution, and a system for expressing large numbers.
Asia Minor (c. 200—100 BC)
An early trigonometric table was apparently said to have been compiled by Hipparchus of Nicaea (180  125 BC).^{[25]} Hipparchus apparently tabulated the corresponding values of arc and chord for a series of angles.^{[25]}^{[26]}
References
 ↑ ^{1.0} ^{1.1} George G. Joseph (2000). The Crest of the Peacock, p. 78. Princeton University Press. ISBN 0691006598.
 ↑ Adams, Winthrope L in Bugh, Glenn Richard. ed. "The Hellenistic Kingdoms". The Cambridge Companion to the Hellenistic World. Cambridge: Cambridge University Press. 2006, p. 39
 ↑ Stanwick, Paul Edmund. Portraits of the Ptolemies: Greek Kings as Egyptian Pharaohs. Austin: University of Texas Press. 2003, p. 23
 ↑ Adams, op cit.
 ↑ Bagnall, R.S. in Susan Walker, ed. Ancient Faces : Mummy Portraits in Roman Egypt (Metropolitan Museum of Art Publications). New York: Routledge, 2000, p. 27
 ↑ ^{6.0} ^{6.1} Bagnall, op cit.
 ↑ Bagnall, pp. 2829
 ↑ qtd. in Alan K. Bowman, Egypt after the Pharaohs, 332 BC − AD 642, Berkeley: University of California Press, 1996, p. 126: "genuine Egyptians can easily be recognized among the linenweavers by their speech."
 ↑ Egyptology Online: Fayoum mummy portraits accessed on January 16, 2007
 ↑ Encyclopædia Britannica Online  Egyptian art and architecture  GrecoRoman Egypt accessed on January 16, 2007
 ↑ Walker, Susan, op cit., p. 24
 ↑ Dentition helps archaeologists to assess biological and ethnic population traits and relationships
 ↑ Irish JD (2006). "Who were the ancient Egyptians? Dental affinities among Neolithic through postdynastic peoples.". Am J Phys Anthropol 129 (4): 52943
 ↑ Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0321016181
 ↑ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html. Retrieved 20080926.
 ↑ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0030295580 p. 141 "No work, except The Bible, has been more widely used... ."
 ↑ Heath, Thomas (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324. http://books.google.com/books?id=LOA5AAAAMAAJ&pg=PR323.
 ↑ ^{18.0} ^{18.1} ^{18.2} ^{18.3} Boyer (1991). "Greek Trigonometry and Mensuration". p. 163. "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue  that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest."
 ↑ Needham, Volume 3, 108.
 ↑ ^{20.0} ^{20.1} ^{20.2} ^{20.3} Boyer (1991). "Greek Trigonometry and Mensuration". pp. 164–166. "The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. [...] Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from A.D. 127 to 151 and, therefore, assume that he was born at the end of the first century. Suidas, a writer who lived in the tenth century, reported that Ptolemy was alive under Marcus Aurelius (emperor from A.D. 161 to 180)."
 ↑ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) Uncertainty about the life of Diophantus is so great that we do not know definitely in which century he lived. Generally he is assumed to have flourished about A.D. 250, but dates a century or more earlier or later are sometimes suggested[...] If this conundrum is historically accurate, Diophantus lived to be eightyfouryears old. [...] The chief Diophantine work known to us is the Arithmetica, a treatise originally in thirteen books, only the first six of which have survived.}"
 ↑ ^{22.0} ^{22.1} (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 180182) "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent."
 ↑ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."
 ↑ O'Connor, J.J. and Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/The_rise_of_calculus.html. Retrieved 20070807.
 ↑ ^{25.0} ^{25.1} Boyer (1991). "Greek Trigonometry and Mensuration". p. 162. "For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted."
 ↑ O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", MacTutor History of Mathematics Archive. (1996).
