The **quadrivium** comprised the four subjects, or arts, taught in medieval universities after the trivium. The word is Latin, meaning “the four ways” or “the four roads”. Together, the trivium and the quadrivium comprised the seven liberal arts.^{[1]} The quadrivium consisted of arithmetic, geometry, music, and astronomy. These followed the preparatory work of the trivium made up of grammar, logic (or dialectic, as it was called at the times), and rhetoric. In turn, the quadrivium was considered preparatory work for the serious study of philosophy and theology.

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## Origins

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.^{[1]} The quadrivium is implicit in early Pythagorean writings and in the *De nuptiis* of Martianus Capella, although the term was not used until Boethius early in the sixth century.^{[2]} 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

^{[3]}.

## Medieval usage

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, and the B.Mus. remains a postgraduate qualification at Oxford and Cambridge universities).

The study was eidetic, approaching the philosophical objectives sought by considering it from each aspect of the quadrivium within the general structure demonstrated by Proclus, namely arithmetic and music on the one hand,^{[4]} and geometry and cosmology on the other.^{[5]}

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.

## Modern usage

In modern applications of the liberal arts as curriculum in colleges or universities, the quadrivium may be considered as 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.^{[6]}

This schema is sometimes referred to as “classical education” but it is more accurately a development of the 12th and 13th centuries 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.

## See also

## References

- ^
^{a}^{b}“Quadrivium“.*New International Encyclopedia*. 1905. **^**Henri Irénée Marrou, “Les Arts Libéreaux 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 0691020906.**^**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