The Quadrivium: An Introduction
Defining the Quadrivium
The quadrivium is the grouping of the four liberal arts of quantity. Preceded by the arts of the trivium, which are the arts of language, they make up the foundation of a liberal education. The quadrivial arts are described by Socrates in Plato’s Republic as the foundation of the education which turns the soul toward being (1). The term quadrivium was developed by Boethius (c. A.D. 480 – 524), who wrote on all the quadrivial arts, as well as incorporating all of them in his Consolation of Philosophy. By the high point of medieval scholasticism, the quadrivium was intended to be studied after the trivium and before natural science, moral philosophy (history, literature, ethics, politics), metaphysical philosophy and theology (2).
The quadrivium can be considered mathematical arts, since they all deal with our understanding of quantity. Arithmetic considers discrete quantity, which quantities are called multitudes. Discrete quantities have no common boundary between one another; they are typically numbers. When we ask ‘How many?’, we are asking about discrete quantity. Geometry, though, considers continuous quantity, which quantities are magnitudes. Continuous quantities have common boundaries between their parts, and examples are surfaces, lines, shapes, time, and space. When we ask ‘How much?’, we are asking about continuous quantity. Both arithmetic and geometry deal with those quantities considered apart from motion (in time or space). Music, however, is the study of discrete quantity (number) in motion (in time), and astronomy is the study of continuous quantity in motion (in time and space).
There is a logical ordering in the quadrivium in which arithmetic and geometry necessarily precede music and astronomy, seeing as we move from what is simpler (quantity considered by itself) to what is more complex (quantity considered in motion). However, there is debate within the Western tradition regarding whether arithmetic or geometry is more fundamental. Nichomachus of Gerasa in his Arithmetic, written in the first century A.D., argued for arithmetic being the first study of the quadrivium. Euclid, however, in his Elements, written in the fourth century B.C., begins with geometry for six books and introduces arithmetic subsequently in books VII and VIII. I would argue that geometry makes a more natural starting place since continuous quantities are more sensible, and thus more quickly knowable, than discrete quantities (numbers), which are abstract. All mathematics entails abstraction, but geometry still utilizes shapes and figures, which are more tangible to the imagination than numbers are. For example, young children can answer ‘Which piece of cake is bigger?’ even before they know how to count. Regardless of where the student begins, the quadrivial arts should be studied together as the student reaches the age of reason, since these mathematical arts all inform and assist one another, as Euclid demonstrates by combining geometry and arithmetic in his Elements.
Music and astronomy follow arithmetic and geometry. Music studies the harmonies that are created through combining musical intervals, which intervals are derived from mathematical ratios in the cosmos. Music considers the production of these intervals through instruments and the ways we combine them in consonance or dissonance, as well as the ways we order and combine those consonances and dissonances to create compositions. The beauty of music is thus inherently mathematical, something that may seem strange or foreign to contemporary, ‘romantic’ sensibilities.
Astronomy studies the movements of the heavenly bodies to discern the regular patterns and to seek to represent their movements in mathematical models. The ancient and medieval understandings of astronomy were primarily mathematical, and they were less concerned with the natural causes of the universe, partly because they had few means to study the heavenly bodies in detail or with accuracy. Nevertheless, they observed predictable, well-ordered patterns in the heavens, and they sought to account for these mathematically through the discipline of astronomy. The modern discipline of astronomy, as pioneered by figures such as Johannes Kepler (1571 – 1630) and Galileo Galilei (1564 – 1642) combines the classical-medieval attempt to yield a mathematically faithful model of the cosmos with the mode of inquiry proper to natural science, which seeks to inquire deeply into the natural causes that produce the phenomena we observe (e.g., understanding the force of gravity which causes the motion of the heavenly bodies). Even though the scope of modern astronomy has broadened and deepened, so to speak, beyond its classical origins, it is impossible to divorce astronomy from an understanding of the mathematical order that undergirds and structures the cosmos as we experience and come to know it.
The Quadrivium as Arts and Sciences
Having summarized the four studies of the quadrivium, we must see that the quadrivial studies are both arts and sciences. They are arts because they are bodies of reasoned, ordered knowledge that enable us to make or do something. For example, geometry allows one to make constructions and proofs, and music allows one to play instrumental music. Nevertheless, they are also sciences. A science is a body of reasoned, ordered knowledge that looks into the nature of a subject for its own sake to yield as certain knowledge as we can have about that subject. A science is speculative, meaning it looks into the nature of something for its own sake. The quadrivial arts are also sciences because, in each of them, we are desiring mathematical knowledge of numbers, magnitudes, musical harmonies, or the movements of heavenly bodies for its own sake. The purpose of the quadrivium is not ultimately utilitarian, as if we were just trying to yield functional geometers, human calculators, musical technicians, or navigational guides. We look into this knowledge because it is inherently worthwhile and because we have an inborn desire to know things. As Aristotle famously says in the opening line of his Metaphysics: “All men by nature desire to know” (3).
That is why these quadrivial arts and sciences are considered to be liberal arts. As liberal arts, their product is not merely the external products they generate (proofs, calculations, compositions, etc.) but a habit of reason within the soul of the learner. Just as grammar, logic, and rhetoric primarily form the mind of the learner in certain habits, so do the quadrivial arts. Their being ‘liberal’ means that they free the human mind in some capacity by enabling it to fulfil its natural function with excellence.
To truly understand this, we must ask: What do the quadrivial arts enable us to do? How do they free us? All the quadrivial (mathematical) arts entail discerning patterns of quantity that exist in the world around us. We perceive information about the world through our senses (sight, hearing, etc.), but it is the human intellect that discerns quantitative patterns in that sensible, material content. Mathematics abstracts the forms of quantity from the actual material things in which we encounter those quantities. For example, when we consider the number ‘two’, we are not considering two trees, or two dogs, but two in itself. This is the work of the human intellect, because the intellect can abstract universal concepts from individual sensible experiences of things in the world.
The quadrivial arts “summon the intellect”, as Socrates describes in Book VII of Plato’sRepublic, to discern what is universal among many particular things, what is equal among things that are otherwise unequal, what is the same among things that are otherwise different, what is unchangeable among things that are otherwise undergoing change (4). Mathematics also allows us to understand what is essential about something from what is merely accidental. Francis Poincaire said that mathematics was the “art of giving the same name to different things”, and this is profoundly insightful (5). Only the human intellect can discern what is the same among things that are otherwise different. The quadrivial arts require us to take sense data and interpret it to reveal what recognizable patterns are in the cosmos around us. Thus the quadrivium allows us to understand the beautiful order of the cosmos around us, for the quadrivium reveals to our intellect that the cosmos actually is well-ordered and governed by mathematical patterns that we can understand. In perfecting and freeing the human intellect, the quadrivium is perfecting that part of us which is proper to us as man, rather than merely as an animal creature.
One of the key things the quadrivium can show us is proportion (Greek analogia). In Book V of his Elements, Euclid defines proportion as the ‘sameness of ratios’ across two different quantities (6). In geometry, two similar triangles are proportionate because the measure of their sides and angles are in the same ratio to one another. More broadly, proportion could be considered the likeness or equality of things that are otherwise unlike or unequal. The quadrivium teaches us to discern proportions, and it opens our mind to recognize the likeness across things that might otherwise be distinct and unlike. If we take seriously that God has created the cosmos according to His own beauty and truth, then why should we not expect to find all kinds of proportions, or even analogies, across the cosmos? The human intellect, unlike the merely sentient animal mind, possesses the ability to discern these proportions and patterns, and thus to recognize that the cosmos is well-ordered, beautiful, and intelligible.
Ascending by the Quadrivium
The quadrivium was traditionally studied and taught as a bridge to higher studies, linking the lowest things with the highest things. Nichomachus of Gerasa, the ancient authority on arithmetic, wrote about the quadrivium this way:
For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to use from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls (7).
As Nichomachus describes, mathematics begins with what is known through the senses but proceeds to what is known only through the intellect. It begins with what is material, mutable, and particular, but through that we come to discern what is immaterial, immutable, and universal. The quadrivium turns the student’s mind to consider what is immaterial, what is universal, what is immutable, and all of these things are more real and more permanent than the transient, changeable things we witness every day. It is an excellent preparation for philosophy and theology, in which we deal with those highest realities that are immaterial and do not exist in matter, but are more real and more permanent than the material things we experience. The quadrivium prepares us ultimately for philosophy and theology, but it was also considered excellent preparation even for natural science and for moral philosophy (history, literature, ethics, politics). The objects of the quadrivial studies – shapes, numbers, musical harmonies, orbits, etc. – are studied and known with greater simplicity and more certainty than the objects of natural science, humanities, philosophy, or theology, all of whose objects are known with more complexity and less certainty. For example, it is easier to understand the properties and perfections of a square than it is to understand justice, a constitution, human nature, or God Himself. The quadrivium trains the intellect in the process of reasoning and discerning universal natures, which skill is employed in all the other studies and disciplines. For example, Thomas Aquinas frequently cites geometrical examples in the Summa Theologiae as an analogous means of helping us understand the ‘highest’ and most complex truths of theology. Thus, the quadrivium naturally leads the student to higher studies and ultimately to the highest study.
The quadrivium should lead us to know and contemplate God Himself when we consider the admirable order, proportion, and harmony He has woven throughout the cosmos. Nicolas Copernicus, a devout Christian as well as revolutionary astronomer, put it thus in his On the Revolutions of Heavenly Spheres: “For who, after applying himself to things which he sees established in the best order and directed by divine ruling, would not through diligent contemplation of them and through a certain habituation be awakened to that which is best and would not wonder at the Artificer of all things, in Whom is all happiness and every good?” (8). These patterns we explore in the quadrivium are ultimately patterns of God Himself, for He is the intelligent designer who built the cosmos and ordered it in such a way that it is intelligible, and who has given us an intellect so that we can understand it. Mathematics is a distinctly human discipline because only humans have intellect, since we are made in God’s image, and thus only humans engage in the process both of abstraction and recognition of patterns which are practiced in mathematics. The quadrivium allows us to fulfill our nature excellently as knowers in God’s creation and it should invoke wonder in us as we consider the order and beauty of the cosmos revealed through the quadrivium.
Works Cited
Plato, The Republic, translated by Allan Bloom (New York: Basic Books, 1968), 521c – d. For a much more detailed treatment of this notion of ‘turning’ the soul by the quadrivium, see Jeffrey Lehman, “The Cave and the Quadrivium: Mathematics in Classical Education” in Principia: A Journal of Classical Education 1, no. 1 (2022): 63 – 74, https://doi.org/10.5840/principia20229275 (accessed January 25, 2025).
See, for example, the order of studies presented by Thomas Aquinas in select places in his writings or that of John of Salisbury in his Metalogicon, I.24. John of Salisbury does not technically lay out an order of studies in the Metalogicon, but he descriptively suggests an order in I.24 which anticipates the fuller, more detailed version of this account of the order of studies which Thomas Aquinas developed and expressed in his writings (see his preface to Super Librum De Causis Expositio and also his commentary on the Ethics, Book VI, Lecture 7). John of Salisbury, Metalogicon, translated and edited by J. B. Hall and KS.B. Keats-Rohan, Corpus Christianorum in translation, Vol. 12 (Turnhout, Belgium: Brepols Publishers, 2013), I.24; Thomas Aquinas, Super Librum De Causis Expositio, translated by Vincent Guagliardo, O.P., Charles Hess, O.P., and Richard Taylor (Washington, D.C.: Catholic University of America Press, 1996), Internet Archive Open Library, https://archive.org/details/commentary-on-the-book-of-causes-aquinas_202106/page/n3/mode/2up (accessed September 8, 2023), Preface; Thomas Aquinas, Super Librum Ethicorum Aristotelis, translated by C. I. Litzinger, O.P. (Chicago: Henry Regnery Company, 1964), https://isidore.co/aquinas/english/Ethics.htm (accessed September 8, 2023), Book VI, Lecture 7.
Aristotle, The Metaphysics, translated by W.D. Ross, The Complete Works of Aristotle: Volume II (Princeton, NJ: Princeton University Press, 1984), 980a.25.
Plato, Republic 523b.
Francis Poincare quoted in Francis Su, Mathematics for Human Flourishing (New Haven, CT: Yale University Press, 2020), 42.
Euclid, Elements, translated by Sir Thomas Heath, Great Books of the Western World, Vol. 11 (Chicago: Encyclopaedia Brittanica, 1952), Book V, Definition 6.
Nichomachus of Gerasa, Introduction to Arithmetic, translated by D’Ooge, Robbins, and Karpinski, in Great Books of the Western World, Vol. 11 (Chicago: Encyclopaedia Brittanica, 1952), Book I, ch. III.6 (p. 812 in this edition).
Nicolaus Copernicus, On the Revolutions of the Heavenly Spheres, translated by Charles Glenn Wallis, in Great Books of the Western World, Vol. 16 (Chicago: Encyclopaedia Brittanica, 1952), Book I (p. 510 in this edition).
