Ancient Greek mathematics
Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the shores of the ancient Mediterranean, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language. The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs is an important difference between Greek mathematics and those of preceding civilizations.
The early history of Greek mathematics is obscure, and traditional narratives of mathematical theorems found before the fifth century BC are regarded as later inventions. It is now generally accepted that treatises of deductive mathematics written in Greek began circulating around the mid-fifth century BC, but the earliest complete work on the subject is Euclid's Elements, written during the Hellenistic period. The works of renown mathematicians Archimedes and Apollonius, as well as of the astronomer Hipparchus, also belong to this period. In the Imperial Roman era, Ptolemy used trigonometry to determine the positions of stars in the sky, while Nicomachus and other ancient philosophers revived ancient number theory and harmonics. During late antiquity, Pappus of Alexandria wrote his Collection, summarizing the work of his predecessors, while Diophantus' Arithmetica dealt with the solution of arithmetic problems by way of pre-modern algebra. Later authors such as Theon of Alexandria, his daughter Hypatia, and Eutocius of Ascalon wrote commentaries on the authors making up the ancient Greek mathematical corpus.
The works of ancient Greek mathematicians were copied in the Byzantine period and translated into Arabic and Latin, where they exerted influence on mathematics in the Islamic world and in Medieval Europe. During the Renaissance, the texts of Euclid, Archimedes, Apollonius, and Pappus in particular went on to influence the development of early modern mathematics. Some problems in Ancient Greek mathematics were solved only in the modern era by mathematicians such as Carl Gauss, and attempts to prove or disprove Euclid's parallel line postulate spurred the development of non-Euclidean geometry. Ancient Greek mathematics was not limited to theoretical works but was also used in other activities, such as business transactions and land mensuration, as evidenced by extant texts where computational procedures and practical considerations took more of a central role.
Etymology
The Greek word derives from , and ultimately from the verb . Strictly speaking, a could be any branch of learning, or anything learnt; however, since antiquity certain were granted special status: arithmetic, geometry, astronomy, and harmonics. These four, which appear listed together around the time of Archytas and Plato, would later become the medieval quadrivium. Geminus of Rhodes later divided the mathematics of the quadrivium in two parts: one dealing with intelligibles, and the other with perceptibles. To the latter, he added mechanics, optics, geodesy, and logistics, which are now part of physics or applied mathematics.Origins
The origins of Greek mathematics are not well understood. The earliest advanced civilizations in Greece were the Minoan and later Mycenaean civilizations, both of which flourished in the second half of the Bronze Age. While these civilizations possessed writing, and many Linear B tablets and similar objects have been deciphered, no mathematical writings have yet been discovered. The mathematics from the preceding Babylonian and Egyptian civilizations were primarily focused on land mensuration and accounting. Although some problems were contrived to be challenging beyond any obvious practical application, there are no signs of explicit theoretical concerns as found in Ancient Greek mathematics. It is generally thought that Babylonian and Egyptian mathematics had an influence on the younger Greek culture, possibly through an oral tradition of mathematical problems over the course of centuries, though no direct evidence of transmission is available.When Greek writing re-emerged in the 7th century BC, following the Late Bronze Age collapse, it was based on an entirely new system derived from the Phoenician alphabet, with Egyptian papyrus being the preferred medium. Because the earliest known mathematical treatises in Greek, starting with Hippocrates of Chios in the 5th century BC, have been lost, the early history of Greek mathematics must be reconstructed from information passed down through later authors, beginning in the mid-4th century BC. Much of the knowledge about early Greek mathematics is thanks to references by Plato, Aristotle, and from quotations of Eudemus of Rhodes' histories of mathematics by later authors. These references provide near-contemporary accounts for many mathematicians active in the 4th century BC. Euclid's Elements is also believed to contain many theorems that are attributed to mathematicians in the preceding centuries.
Archaic period
Ancient Greek tradition attributes the origin of Greek mathematics to either Thales of Miletus, one of the legendary Seven Sages of Greece, or to Pythagoras of Samos, both of whom supposedly visited Egypt and Babylon and learned mathematics there. However, modern scholarship tends to be skeptical of such claims as neither Thales or Pythagoras left any writings that were available in the Classical period. Additionally, widespread literacy and the scribal culture that would have supported the transmission of mathematical treatises did not emerge fully until the 5th century; the oral literature of their time was primarily focused on public speeches and recitations of poetry. The standard view among historians is that the discoveries Thales and Pythagoras are credited with, such as Thales' Theorem, the Pythagorean theorem, and the Platonic solids, are the product of attributions by much later authors.Classical Greece
The earliest traces of Greek mathematical treatises appear in the second half of the fifth century BC. According to Eudemus, Hippocrates of Chios was the first to write a book of Elements in the tradition later continued by Euclid. Fragments from another treatise written by Hippocrates on lunes also survives, possibly as an attempt to square the circle. Eudemus' states that Hippocrates studied with an astronomer named Oenopides of Chios. Other mathematicians associated with Chios include Andron and Zenodotus, who may be associated with a "school of Oenopides" mentioned by Proclus.Although many stories of the early Pythagoreans are likely apocryphal, including stories about people being drowned or exiled for sharing mathematical discoveries, some fifth-century Pythagoreans may have contributed to mathematics. Beginning with Philolaus of Croton, a contemporary of Socrates, studies in arithmetic, geometry, astronomy, and harmonics became increasingly associated with Pythagoreanism. Fragments of Philolaus' work are preserved in quotations from later authors. Aristotle is one of the earliest authors to associate Pythagoreanism with mathematics, though he never attributed anything specifically to Pythagoras.
Other extant evidence shows fifth-century philosophers' acquaintance with mathematics: Antiphon claimed to be able to construct a rectilinear figure with the same area as a given circle, while Hippias is credited with a method for squaring a circle with a neusis construction. Protagoras and Democritus debated the possibility for a line to intersect a circle at a single point. According to Archimedes, Democritus also asserted, apparently without proof, that the area of a cone was 1/3 the area of a cylinder with the same base, a result which was later proved by Eudoxus of Cnidus.
Mathematics in the time of Plato
While Plato was not a mathematician, numerous early mathematicians were associated with Plato or with his Academy. Familiarity with mathematicians' work is also reflected in several Platonic dialogues were mathematics are mentioned, including the Meno, the Theaetetus, the Republic, and the Timaeus.Archytas, a Pythagorean philosopher from Tarentum, was a friend of Plato who made several contributions to mathematics, including solving the problem of doubling the cube, now known to be impossible with only a compass and a straightedge, using an alternative method. He also systematized the study of means, and possibly worked on optics and mechanics. Archytas has been credited with early material found in Books VII–IX of the Elements, which deal with elementary number theory.
Theaetetus is one of the main characters in the Platonic dialogue named after him, where he works on a problem given to him by Theodorus of Cyrene to demonstrate that the square roots of several numbers from 3 to 17 are irrational, leading to the construction now known as the Spiral of Theodorus. Theaetetus is traditionally credited with much of the work contained in Book X of the Elements, concerned with incommensurable magnitudes, and Book XIII, which outlines the construction of the regular polyhedra. Although some of the regular polyhedra were certainly known previously, he is credited with their systematic study and the proof that only five of them exist.
Another mathematician who might have visited Plato's Academy is Eudoxus of Cnidus, associated with the theory of proportion found in Book V of the Elements. Archimedes credits Eudoxus with a proof that the volume of a cone is one-third the volume of a cylinder with the same base, which appears in two propositions in Book XII of the Elements. He also developed an astronomical calendar, now lost, that remains partially preserved in Aratus' poem Phaenomena. Eudoxus seems to have founded a school of mathematics in Cyzicus, where one of Eudoxus' students, Menaechmus, went on to develop a theory of conic sections.