Hellenistic mathematics

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), which in turn influenced medieval European 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.

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)

Main article: Egyptian mathematics

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.

Most of the mathematical texts written in Greek have been found in Egypt, as well as Greece, Asia Minor, Mesopotamia, and Sicily.

Demographics

Fayum mummy portraits of Greco-Egyptians in the 1st century CE.

Mummy portrait of Greco-Egyptian in the 1st century CE.

Mummy portrait of Eutyches, a Greco-Egyptian in the 2nd century CE.

Under Greco-Roman 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. Faiyum's earliest Greek inhabitants were soldier-veterans and cleruchs (elite military officials) who were settled by the Ptolemaic kings on reclaimed lands. 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. 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. By the Roman period, much of the "Greek" population of Faiyum was made-up of either Hellenized Egyptians or people of mixed Egyptian-Greek origins. 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."

While commonly believed to represent Greek settlers in Egypt, the Faiyum mummy portraits instead reflect the complex synthesis of the predominant Egyptian culture and that of the elite Greek minority in the city. 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 self-perception of being Greek. The dental morphology of the Roman-period 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.

The mathematics historian Victor J. Katz has criticized the modern misconception portraying the ancient Hellenistic scholars of Egypt as ethnically Greek or European, writing:

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)

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus, Egypt, and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.

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. 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. 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. 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. Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°. Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus". He further gave his famous "rule of six quantities".

Problem II.8 in Diophantus' Arithmetica (1670 edition), annotated with Fermat's comment which became Fermat's Last Theorem.

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. 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 sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle formula $\sin^2({x/2}) = \frac{1 - \cos(x)}{2}$. Ptolemy used these results to create trigonometric tables.

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. 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. In Arithmetica, Diophantus used symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; 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.

Hellenistic Sicily (c. 300—200 BC)

See also: Greek mathematics and Science and technology in Italy

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. 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)

See also: Greek mathematics and Babylonian mathematics

An early trigonometric table was apparently said to have been compiled by Hipparchus of Nicaea (180 - 125 BC). Hipparchus apparently tabulated the corresponding values of arc and chord for a series of angles.