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The quadrivium (plural: quadrivia) is the four subjects, or arts, taught after teaching the trivium. The word is Latin, meaning "the four ways" (or a "place where four roads meet"), and its use for the four subjects has been attributed to Boethius or Cassiodorus in the 6th century. Together, the trivium and the quadrivium comprised the seven liberal arts (based on thinking skills), as opposed to the practical arts (such as medicine and architecture).
The quadrivium consisted of arithmetic, geometry, music and astronomy. These followed the preparatory work of the trivium made up of grammar, logic and rhetoric. In turn, the quadrivium was considered preparatory work for the serious study of philosophy (sometimes called the "liberal art par excellence") and theology.
These four studies compose the secondary part of the curriculum outlined by Plato in The Republic, and are described in the seventh book of that work (in the order Arithmetic, Geometry, Astronomy, Music.) The quadrivium is implicit in early Pythagorean writings and in the De nuptiis of Martianus Capella, although the term "quadrivium" was not used until Boethius early in the sixth century. As Proclus wrote:
The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.
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