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Quadrivium
From the time of Plato through the Middle Ages, the quadrivium (plural: quadrivia, Latin for "four ways") was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the trivium, consisting of grammar, logic, and rhetoric. Together, the trivium and the quadrivium comprised the seven liberal arts, and formed the basis of a liberal arts education in Western society until gradually displaced as a curricular structure by the studia humanitatis and its later offshoots, beginning with Petrarch in the 14th century. The seven classical arts were considered "thinking skills" and were distinguished from practical arts, such as medicine and architecture.
The four mathematical arts were recognized by Pythagoreans such as Nicomachus of Gerasa, but the use of quadrivium as a term for these four subjects has been attributed to Boethius, when he affirmed that the height of philosophy can be attained only following "a sort of fourfold path" (quodam quasi quadruvio). It was considered the foundation for the study of philosophy (sometimes called the "liberal art par excellence") and theology. The quadrivium was the upper division of medieval educational provision in the liberal arts, which comprised arithmetic (number in the abstract), geometry (number in space), music (number in time), and astronomy (number in space and time).
Educationally, the trivium and the quadrivium imparted to the student the seven essential thinking skills of classical antiquity. Altogether the Seven Liberal Arts belonged to the so-called 'lower faculty' (of Arts), whereas Medicine, Jurisprudence (Law), and Theology were established in the three so-called 'higher' faculties. It was therefore quite common in the middle ages for lecturers in the lower trivium and/or quadrivium faculty to be students themselves in one of the higher faculties. Philosophy was typically neither a subject nor a faculty in its own right, but was rather present implicitly as an 'auxiliary tool' within the discourses of the higher faculties, especially theology; the separation of philosophy from theology and its elevation to an autonomous academic discipline were post-medieval developments.
Displacement of the quadrivium by other curricular approaches from the time of Petrarch gained momentum with the subsequent Renaissance emphasis on what became the modern humanities, one of four liberal arts of the modern era, alongside natural science (where much of the actual subject matter of the original quadrivium now resides), social science, and the arts; though it may appear that music in the quadrivium would be a modern branch of performing arts, it was then an abstract system of proportions that was carefully studied at a distance from actual musical practice, and effectively a branch of music theory more tightly bound to arithmetic than to musical expression.[citation needed]
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.
At many medieval universities, this would have been the course leading to the degree of Master of Arts (after the BA).[citation needed] 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.
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Quadrivium
From the time of Plato through the Middle Ages, the quadrivium (plural: quadrivia, Latin for "four ways") was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the trivium, consisting of grammar, logic, and rhetoric. Together, the trivium and the quadrivium comprised the seven liberal arts, and formed the basis of a liberal arts education in Western society until gradually displaced as a curricular structure by the studia humanitatis and its later offshoots, beginning with Petrarch in the 14th century. The seven classical arts were considered "thinking skills" and were distinguished from practical arts, such as medicine and architecture.
The four mathematical arts were recognized by Pythagoreans such as Nicomachus of Gerasa, but the use of quadrivium as a term for these four subjects has been attributed to Boethius, when he affirmed that the height of philosophy can be attained only following "a sort of fourfold path" (quodam quasi quadruvio). It was considered the foundation for the study of philosophy (sometimes called the "liberal art par excellence") and theology. The quadrivium was the upper division of medieval educational provision in the liberal arts, which comprised arithmetic (number in the abstract), geometry (number in space), music (number in time), and astronomy (number in space and time).
Educationally, the trivium and the quadrivium imparted to the student the seven essential thinking skills of classical antiquity. Altogether the Seven Liberal Arts belonged to the so-called 'lower faculty' (of Arts), whereas Medicine, Jurisprudence (Law), and Theology were established in the three so-called 'higher' faculties. It was therefore quite common in the middle ages for lecturers in the lower trivium and/or quadrivium faculty to be students themselves in one of the higher faculties. Philosophy was typically neither a subject nor a faculty in its own right, but was rather present implicitly as an 'auxiliary tool' within the discourses of the higher faculties, especially theology; the separation of philosophy from theology and its elevation to an autonomous academic discipline were post-medieval developments.
Displacement of the quadrivium by other curricular approaches from the time of Petrarch gained momentum with the subsequent Renaissance emphasis on what became the modern humanities, one of four liberal arts of the modern era, alongside natural science (where much of the actual subject matter of the original quadrivium now resides), social science, and the arts; though it may appear that music in the quadrivium would be a modern branch of performing arts, it was then an abstract system of proportions that was carefully studied at a distance from actual musical practice, and effectively a branch of music theory more tightly bound to arithmetic than to musical expression.[citation needed]
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.
At many medieval universities, this would have been the course leading to the degree of Master of Arts (after the BA).[citation needed] 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.