The quadrivium comprised the four subjects, or arts, taught in medieval universities
A university is an institution of higher education and research, which grants academic degrees in a variety of subjects. A university is an organisation that provides both undergraduate education and postgraduate education...

, after teaching the trivium. The word is Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

, meaning "the four ways" (or a "place where four roads meet"), and its use for the 4 subjects has been attributed to Boethius or Cassiodorus
Flavius Magnus Aurelius Cassiodorus Senator , commonly known as Cassiodorus, was a Roman statesman and writer, serving in the administration of Theodoric the Great, king of the Ostrogoths. Senator was part of his surname, not his rank.- Life :Cassiodorus was born at Scylletium, near Catanzaro in...

 in the 6th century. Together, the trivium and the quadrivium comprised the seven liberal arts
Liberal arts
The term liberal arts refers to those subjects which in classical antiquity were considered essential for a free citizen to study. Grammar, Rhetoric and Logic were the core liberal arts. In medieval times these subjects were extended to include mathematics, geometry, music and astronomy...

 (based on thinking skills), as opposed to the practical arts (such as medicine and architecture). The quadrivium consisted of arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, and astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

. These followed the preparatory work of the trivium made up of grammar
In linguistics, grammar is the set of structural rules that govern the composition of clauses, phrases, and words in any given natural language. The term refers also to the study of such rules, and this field includes morphology, syntax, and phonology, often complemented by phonetics, semantics,...

, logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, and rhetoric
Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...

. In turn, the quadrivium was considered preparatory work for the serious study of philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

 (sometimes called the "liberal art par excellence") and theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

. The word "trivia" has been rarely used to refer to the trivium.


These four studies compose the secondary part of the curriculum outlined by Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

 in The Republic, and are described in the seventh book of that work.
The quadrivium is implicit in early Pythagorean
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...

 writings and in the De nuptiis of Martianus Capella
Martianus Capella
Martianus Minneus Felix Capella was a pagan writer of Late Antiquity, one of the earliest developers of the system of the seven liberal arts that structured early medieval education...

, although the term "quadrivium" was not used until Boethius early in the sixth century. As Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...


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.

Medieval usage

At many medieval universities, this would have been the course leading to the degree of Master of Arts (after the BA
Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...

). 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
Bachelor of Philosophy
Bachelor of Philosophy is the title of an academic degree. The degree usually involves considerable research, either through a thesis or supervised research projects...

 and B.Litt.
British degree abbreviations
Degree abbreviations are used as an alternative way to specify an academic degree instead of spelling out the title in full, such as in reference books like Who's Who and on business cards...

 degrees are examples in the field of philosophy, and the B.Mus.
Bachelor of Music
Bachelor of Music is an academic degree awarded by a college, university, or conservatory upon completion of program of study in music. In the United States, it is a professional degree; the majority of work consists of prescribed music courses and study in applied music, usually requiring a...

 remains a postgraduate qualification at Oxford
University of Oxford
The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...

 and Cambridge universities
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...


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, and geometry and cosmology on the other.

The subject of music within the quadrivium was originally the classical subject of harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

s, in particular the study of the proportions between the music intervals created by the division of a monochord
A monochord is an ancient musical and scientific laboratory instrument. The word "monochord" comes from the Greek and means literally "one string." A misconception of the term lies within its name. Often a monochord has more than one string, most of the time two, one open string and a second string...

. 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
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

 and its relationship to physical space or time: arithmetic was pure number, geometry was number in space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, music number in time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, and astronomy number in space and time
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. Morris Kline
Morris Kline
Morris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.Kline grew up in Brooklyn and in Jamaica, Queens...

 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 of the 12th century
The Renaissance of the 12th century was a period of many changes at the outset of the High Middle Ages. It included social, political and economic transformations, and an intellectual revitalization of Western Europe with strong philosophical and scientific roots...

 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
Classical education movement
The Classical education movement advocates a form of education based in the traditions of Western culture, with a particular focus on education as understood and taught in the Middle Ages. The curricula and pedagogy of classical education was first developed during the Middle Ages by Martianus...


See also

  • Trivium
  • Andreas Capellanus
    Andreas Capellanus
    Andreas Capellanus was the 12th-century author of a treatise commonly known as De amore , and often known in English, somewhat misleadingly, as The Art of Courtly Love, though its realistic, somewhat cynical tone suggests that it is in some measure an antidote to courtly love...

  • The Classical education movement
    Classical education movement
    The Classical education movement advocates a form of education based in the traditions of Western culture, with a particular focus on education as understood and taught in the Middle Ages. The curricula and pedagogy of classical education was first developed during the Middle Ages by Martianus...

  • Degrees of the University of Oxford
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