Ancient Greek mathematics

Ancient Greek mathematics refers to the historical development of mathematical ideas and texts in Ancient Greece during Classical and Late antiquity, mostly from the 5th century BC to the 6th century AD.^[1][2] Greek mathematicians lived in cities spread around the shores of the Mediterranean, from Anatolia to Italy and North Africa, but were united by Hellenistic culture and the Ancient Greek language.^[3] 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, such as Ancient Egypt and Babylonia.^[4][5]

The early history of Greek mathematics is obscure; traditional ascriptions of basic mathematical theorems to legendary sages such as Thales or Pythagoras are much later inventions. By the beginning of the 5th century BC, there are traces of written treatises on mathematics, which developed further in the 4th century BC at institutions such as the Platonic academy. These early mathematical discoveries were compiled in the Hellenistic period by Euclid of Alexandria at the beginning of the third century in his Elements, our earliest complete text on the subject that is now referred to as Euclidean geometry. The 3rd century BC saw further developments in geometry and mechanics by Archimedes, many of whose writings survive, and the development of a theory of conic sections as preserved in the works of Apollonius of Perga. Ancient Greek mathematics encompassed not only on disciplines traditionally included in modern mathematics, such as geometry and arithmetic, but also astronomy and music. Ancient Greek astronomers such as Hipparchus and Ptolemy developed trigonometry to determine the positions of stars in the sky, while Nicomachus and other ancient music theorists developed a theory of harmonics. In the Roman period, Diophantus Arithmetica outlined a theory for the solution of Diophantine equations that would later be developed in the medieval Islamic world into algebra. In the 4th century, Pappus of Alexandria wrote his Collection, summarizing the work of his predecessors, much of which is now lost, documenting the history of attempts to solve problems such as squaring the circle, angle trisection, or doubling the cube using only a straight edge and compass, Theon of Alexandria, and his daughter Hypatia wrote commentaries on the works of earlier mathematicians, including Ptolemy and Diophantus. Later commentators in the 6th century such as Eutocius of Ascalon also wrote commentaries on the works of Apollonius and Archimedes.

The works of Ancient Greek mathematics were copied in the medieval 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 went on to influence the development of early modern mathematicians including Fermat and Descartes, who created number theory and analytic geometry based on their studies of Greek mathematical texts. Many of the problems of Ancient Greek mathematics were only solved in the modern era, by mathematicians such as Gauss. Attempts to derive geometry without Euclid's parallel line postulate spurred the development of non-Euclidean geometry.