Philolaus in the context of "Nicomachus"

Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Modern scholars disagree regarding Pythagoras's education and influences, but most agree that he travelled to Croton in southern Italy around 530 BC, where he founded a school in which initiates were allegedly sworn to secrecy and lived a communal, ascetic lifestyle.

In antiquity, Pythagoras was credited with mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, the sphericity of the Earth, the identity of the morning and evening stars as the planet Venus, and the division of the globe into five climatic zones. He was reputedly the first man to call himself a philosopher ("lover of wisdom"). Historians debate whether Pythagoras made these discoveries and pronouncements, as some of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors, such as Hippasus and Philolaus.

Timaeus (/taɪˈmiːəs/; Ancient Greek: Τίμαιος, romanized: Timaios, pronounced [tǐːmai̯os]) is one of Plato's dialogues, mostly in the form of long monologues given by Critias and Timaeus, written c. 360 BC. The work puts forward reasoning on the possible nature of the physical world and human beings and is followed by the dialogue Critias.

Participants in the dialogue include Socrates, Timaeus, Hermocrates, and Critias. Some scholars believe that it is not the Critias of the Thirty Tyrants who appears in this dialogue, but his grandfather, also named Critias. At the beginning of the dialogue, the absence of another, unknown dialogue participant, present on the day before, is bemoaned. It has been suggested from some traditions—Diogenes Laertius (VIII 85) from Hermippus of Smyrna (3rd century BC) and Timon of Phlius (c. 320 – c. 235 BC)—that Timaeus was influenced by a book about Pythagoras, written by Philolaus, although this assertion is generally considered false.

Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory": The first is the "rudiments", that are needed to understand music notation (key signatures, time signatures, and rhythmic notation); the second is learning scholars' views on music from antiquity to the present; the third is a sub-topic of musicology that "seeks to define processes and general principles in music". The musicological approach to theory differs from music analysis "in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built."

Music theory is frequently concerned with describing how musicians and composers make music, including tuning systems and composition methods among other topics. Because of the ever-expanding conception of what constitutes music, a more inclusive definition could be the consideration of any sonic phenomena, including silence. This is not an absolute guideline, however; for example, the study of "music" in the Quadrivium liberal arts university curriculum, that was common in medieval Europe, was an abstract system of proportions that was carefully studied at a distance from actual musical practice. But this medieval discipline became the basis for tuning systems in later centuries and is generally included in modern scholarship on the history of music theory.

Heliocentrism (also known as the heliocentric model) is a superseded astronomical model in which Earth and planets orbit around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed Earth at the center. The notion that Earth revolves around the Sun had been proposed as early as the 3rd century BC by Aristarchus of Samos, who had been influenced by a concept presented by Philolaus of Croton (c. 470 – 385 BC). In the 5th century BC the Greek philosophers Philolaus and Hicetas had the thought on different occasions that Earth was spherical and revolving around a "mystical" central fire, and that this fire regulated the universe. In medieval Europe, however, Aristarchus' heliocentrism attracted little attention—possibly because of the loss of scientific works of the Hellenistic period.

It was not until the 16th century that a mathematical model of a heliocentric system was presented by the Renaissance mathematician, astronomer, and Catholic cleric, Nicolaus Copernicus, leading to the Copernican Revolution. In the following century, Johannes Kepler introduced elliptical orbits, and Galileo Galilei presented supporting observations made using a telescope.

An astronomical system positing that the Earth, Moon, Sun, and planets revolve around an unseen "Central Fire" was developed in the fifth century BC and has been attributed to the Pythagorean philosopher Philolaus. The system has been called "the first coherent system in which celestial bodies move in circles", anticipating Copernicus in moving "the earth from the center of the cosmos [and] making it a planet". Although its concepts of a Central Fire distinct from the Sun, and a nonexistent "Counter-Earth" were erroneous, the system contained the insight that "the apparent motion of the heavenly bodies" was (in large part) due to "the real motion of the observer". How much of the system was intended to explain observed phenomena and how much was based on myth, mysticism, and religion is disputed. While the departure from traditional reasoning is impressive, other than the inclusion of the five visible planets, very little of the Pythagorean system is based on genuine observation. In retrospect, Philolaus's views are "less like scientific astronomy than like symbolical speculation."