The quadrivium (plural: quadrivia) is the four subjects, or arts, taught after teaching the trivium. The word is Latin, meaning four ways 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.
At many medieval universities, this would have been the course leading to the degree of Master of Arts (after the BA). After the MA, the student could enter for bachelor's degrees of the higher faculties (Theology, Medicine or Law). To this day, some of the postgraduate degree courses lead to the degree of Bachelor (the B.Phil and B.Litt. degrees are examples in the field of philosophy).
The study was eclectic, approaching the philosophical objectives sought by considering it from each aspect of the quadrivium within the general structure demonstrated by Proclus (412–485 AD), namely arithmetic and music on the one hand, and geometry and cosmology on the other.
The subject of music within the quadrivium was originally the classical subject of harmonics, in particular the study of the proportions between the music intervals created by the division of a monochord. A relationship to music as actually practised was not part of this study, but the framework of classical harmonics would substantially influence the content and structure of music theory as practised both in European and Islamic cultures.
In modern applications of the liberal arts as curriculum in colleges or universities, the quadrivium may be considered to be the study of number and its relationship to physical space or time: arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time. Morris Kline classifies the four elements of the quadrivium as pure (arithmetic), stationary (geometry), moving (astronomy) and applied (music) number.
This schema is sometimes referred to as "classical education" but it is more accurately a development of the 12th and 13th centuries Renaissance with recovered classical elements, rather than an organic growth from the educational systems of antiquity. The term continues to be used by the Classical education movement and at the independent Oundle School, in the United Kingdom.
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- Kohler, Kaufmann. "Wisdom". Jewish Encyclopedia. Retrieved 2015-11-07.
- "Part I: The Age of Augustine", ND.edu, 2010, webpage: ND205.
- "quadrivium (education)", Britannica Online, 2011, web: EB.
- Gilman, D. C.; Thurston, H. T.; Colby, F. M., eds. (1905). "Quadrivium". New International Encyclopedia (1st ed.). New York: Dodd, Mead.
- Daniel Coit Gilman et al. (1905) New International Encyclopedia, lemma "Arts, Liberal"
- Henri-Irénée Marrou, "Les Arts Libéraux dans l'Antiquité Classique", pp. 6-27 in Arts Libéraux et Philosophie au Moyen Âge, (Paris: Vrin / Montréal: Institut d'Études Médiévales), 1969, pp. 18-19.
- Proclus, A commentary on the first book of Euclid's Elements, xii, trans. Glenn Raymond Morrow (Princeton: Princeton University Press) 1992, pp. 29-30. ISBN 0-691-02090-6.
- Craig Wright, The Maze and the Warrior - Symbols in Architecture, Theology, and Music, Harvard University Press 2001
- Laura Ackerman Smoller, History, Prophecy and the Stars: Christian Astrology of Pierre D'Ailly, 1350-1420, Princeton University Press 1994
- Morris Kline, "The Sine of G Major", Mathematics in Western Culture, Oxford University Press 1953
- "Oundle School - Improving Intellectual Challenge". The Boarding Schools' Association. 27 October 2014.
Each of these iterations have recently been discussed in a conference at King's College London on "The Future of Liberal Arts" at schools and universities.