Nicomachus in the context of "Philolaus"

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.

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).It is an example of an algorithm, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity.

From the time of Plato through the Middle Ages, the quadrivium (plural: quadrivia, Latin for "four ways") was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the trivium, consisting of grammar, logic, and rhetoric. Together, the trivium and the quadrivium comprised the seven liberal arts, and formed the basis of a liberal arts education in Western society until gradually displaced as a curricular structure by the studia humanitatis and its later offshoots, beginning with Petrarch in the 14th century. The seven classical arts were considered "thinking skills" and were distinguished from practical arts, such as medicine and architecture.

The four mathematical arts were recognized by Pythagoreans such as Nicomachus of Gerasa, but the use of quadrivium as a term for these four subjects has been attributed to Boethius, when he affirmed that the height of philosophy can be attained only following "a sort of fourfold path" (quodam quasi quadruvio). It was considered the foundation for the study of philosophy (sometimes called the "liberal art par excellence") and theology. The quadrivium was the upper division of medieval educational provision in the liberal arts, which comprised arithmetic (absolute number), music (relative number), geometry (magnitude at rest), and astronomy (magnitude in motion).

Asclepius of Tralles (Greek: Ἀσκληπιός; died c. 560–570) was a student of Ammonius Hermiae. Two works of his survive:

• Commentary on Aristotle's Metaphysics, books I-VII (In Aristotelis metaphysicorum libros Α - Ζ (1 - 7) commentaria, ed. Michael Hayduck, Commentaria in Aristotelem Graeca, VI.2, Berin: Reiner, 1888).
• Commentary on Nicomachus' Introduction to Arithmetic (Leonardo Tarán, Asclepius of Tralles, Commentary to Nicomachus' Introduction to Arithmetic, Transactions of the American Philosophical Society (n.s.), 59: 4. Philadelphia, 1969.

Both works seem to be notes on the lectures conducted by Ammonius.