theory more tightly bound to arithmetic than to musical expression. 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 (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 studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving. 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). 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 (AD 412–485), 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 musical 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 in both European and Islamic cultures. Modern usage 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 space or time: arithmetic was pure number, geometry was number in space, music was number in time, and astronomy was number in space and time. Morris Kline classified the four elements of the quadrivium as