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.
