The Quadrivium is the second part of the seven classical liberal arts. Whereas the three arts of Grammar, Logic, and Rhetoric make up the Trivium, the final four arts—Arithmetic, Geometry, Music, and Astronomy—make up the Quadrivium, and together these form the complete classical liberal arts curriculum. The arts of the Quadrivium are the ancient mathematical arts, ordered to the study of Quantity, which is the second of the ten Categories of being identified by Aristotle. These arts were studied throughout history to prepare students for the study of philosophy and theology.
In modern schools, mathematics is studied for practical purposes. Students are taught only what is needed for business, science, or college admission. The goal is mere utility. The classical study of the Quadrivium is entirely different. It is not ordered to utility but to the preparation of the intellect for the contemplation of truth. The Quadrivium does not train students for work but enables the mind to rise from knowledge of physical things to the understanding of immaterial and eternal realities.
The modern “classical education movement” does not teach the Quadrivium. The four mathematical arts are either omitted entirely or replaced with modern math courses. This omission reveals that such programs are not grounded in the true liberal arts tradition and are not rightly ordered to the study of philosophy and theology. Classical philosophy and scholastic theology assume that a student has already studied the seven liberal arts. A student who has not studied the true Quadrivium is not prepared to advance to higher studies.
Aristotle explained that there are three different kinds of things studied by wise men. Material or “physical” things are studied by means of the senses. Immaterial or “metaphysical” things are studied by reason alone. Mathematical things exist between these and are studied by both the senses and reason. The four arts of the Quadrivium each study a different kind of mathematical reality. Quantity divides into two species: Multitude and Magnitude. Multitude refers to a quantity that can be divided into separate units. Magnitude refers to a continuous quantity that can be divided into parts.
Arithmetic is the study of absolute multitude. It is studied in the Classical Liberal Arts Academy using the Introduction to Arithmetic by Nicomachus of Gerasa, an ancient text used by the early Church and preserved in the schools of the Middle Ages. This text teaches the science of number as it was understood in the Pythagorean tradition.
Music is the study of relative multitude. It concerns number in relation and proportion, particularly in sound. It is studied in the Academy through the medieval work On Music by Boethius, a Christian philosopher whose writings shaped the study of music throughout Church history.
Geometry is the study of magnitude at rest. It investigates the properties and relations of points, lines, surfaces, and solids. It is studied in the Academy through the Elements of Euclid, the standard geometry text used throughout classical and medieval education.
Astronomy is the study of magnitude in motion. It concerns the orderly movements of the heavens. It is studied in the Academy through the Almagest by the ancient astronomer Ptolemy and Aristotle’s On the Heavens. These texts present the science of the cosmos as it was understood in the ancient and medieval world.
The Quadrivium prepares the student to rise from the perception of material things to the understanding of immaterial realities. The mind first perceives the world through the senses, then begins to understand quantity through mathematics, and is finally prepared to contemplate being itself in philosophy and divine truth in theology. A student who does not study the Quadrivium lacks the intellectual training needed to make this ascent.
The Classical Liberal Arts Academy teaches the full Quadrivium through its original sources and methods. These arts are not replaced, adapted, or reduced. They are studied as they were studied by the philosophers, saints, and doctors of the Catholic tradition.
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy