The Order of the Seven Liberal Arts: Substance, Quantity and the Mind

The seven liberal arts can be practised one at a time, but the tradition also claims that they form an ordered whole. This page sets out one classical account of that order, separating ancient evidence from later interpretation. The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium covers how to practise each art. Translations are loose unless a translator is named.

Why "liberal"

Aristotle draws the line first. Any occupation that makes the body or mind of a free person less fit for virtue is banausic, a servile trade (Politics VIII.2, 1337b), while wisdom is "the only free science" because it alone exists for its own sake (Metaphysics I.2, 982b25-28). Seneca rejects any study whose end is money-making: liberal studies are so called because they are worthy of a free person (liber), though only wisdom, which makes a person free, is truly liberal (Epistle 88.1-2). Established as history. The ancient distinction assumed that slaves did the servile work. Without that assumption, what survives is a difference of purpose: study for understanding, not for gain.

An art is a habit of the mind

Classically, an art (techne, ars) is not a skill set. Aristotle calls it a hexis, a settled state concerned with making and joined with true reasoning (Nicomachean Ethics VI.4, 1140a), and such states come from doing: "men become builders by building" (II.1, 1103a31-b2, trans. Ross). Aquinas puts it in Latin as recta ratio factibilium, right reason about things to be made (Summa Theologiae I-II, q. 57, aa. 3-4). Aristotle's arts produce things, while arithmetic is theoretical, so Aquinas adds that speculative reason has works too, such as a syllogism, a speech, a count or a measurement. The habits ordered to them are arts "by a kind of comparison", and liberal as opposed to the arts of bodily work (a. 3, ad 3).

Two things follow for this site. A habit comes from repeated, corrected practice, so an art cannot be had by reading about it; hence every page ends with something to do. And the habit includes knowing why a method works; hence the site asks for proof (Proof and Precise Reasoning: From Arguments to Theorems), not a method that merely seems to work.

One reading of the order: substance and quantity

Aristotle sorts what can be said of things into ten categories, among them substance (what a thing is) and quantity (how much) (Categories 4-6). Substance is prior to the rest in definition, in knowledge and in time (Metaphysics VII.1, 1028a31-b2).

One scholastic reading builds on this: the trivium serves the intellect as it apprehends, judges and expresses things, which it first knows as substances; the quadrivium studies quantity, the measurable side of things. This is an interpretation. Aristotle never divides the arts by the categories and never lists seven. Aquinas, citing Hugh of St Victor, is more modest: the seven do not adequately divide theoretical philosophy, but are the studies in which beginners in philosophy were first trained (Commentary on Boethius's De Trinitate, q. 5, a. 1, ad 3).

The stronger claim, that the division mirrors reality, belongs to twentieth-century neo-scholastic teaching, of which Sister Miriam Joseph's The Trivium (1937) is a well-known example. Even her version differs: the trivium pertains to mind and the quadrivium to matter, whose "outstanding characteristic" is quantity, and her grammar sorts words by the categories, nouns standing for substance (chapters 1 to 3). "Trivium for substance" goes a step further.

The trivium: apprehending, judging, expressing

Grammar is "the science of speaking correctly and the interpretation of the poets" (recte loquendi scientiam et poetarum enarrationem, Quintilian, Institutio Oratoria I.4.2). Rhetoric is "the science of speaking well" (bene dicendi scientia, II.15.34; cf. II.17.37), which, Quintilian adds, takes in the speaker's character, since no one can speak well who is not good. Logic, for Aquinas, is the art directing the act of reason itself, "so that it proceeds in order, easily and without error" (Commentary on the Posterior Analytics, proemium). Established as history.

Aquinas takes two acts of the intellect from Aristotle (De Anima III.6), grasping what a thing is and composing or dividing (judgment), and adds a third, reasoning from the known to the unknown (Commentary on the Peri Hermeneias I, lect. 1). A popular modern scheme gives grammar the first act, logic judgment and reasoning, rhetoric expression. That mapping is an interpretation, not Aquinas's: for him logic itself treats all three acts, in the Categories, On Interpretation and the Analytics (Posterior Analytics proemium). Loosely, the three arts run from thought to speech: say it correctly, test it, put it to others. See How to Read a Great Book: Levels, Modes and the Completion Standard for grammar as slow reading and English Expression Mastery: Words, Sentences, Paragraphs and Speech for rhetoric.

The quadrivium: quantity in four modes

Aristotle divides quantity into discrete, such as number, and continuous, such as lines, surfaces and solids (Categories 6, 4b20). A plurality (multitude) is counted; a magnitude is measured (Metaphysics V.13, 1020a7-11). Nicomachus of Gerasa divides each again (Introduction to Arithmetic I.3; Proclus gives the same Pythagorean scheme, Commentary on Euclid, Prologue I). Multitude in itself (even, odd) is arithmetic; multitude in relation (double, half) is music; magnitude at rest is geometry; magnitude in motion is sphairike, "sphaeric", which D'Ooge translates as astronomy. Boethius, adapting Nicomachus, keeps the order arithmetic, music, geometry, astronomy and calls the four a quadruvium, the road by which a more excellent mind is led from the senses to the surer things of understanding (De institutione arithmetica I.1). Established as history.

Music here means harmony grounded in ratio, not performance. Boethius ranks performers, composers and judges, and only the judge, who weighs music "according to speculation or reason", is a musicus (De institutione musica I.34, trans. Bower). He also denies the senses the whole judgment, though the discipline begins in hearing (I.9). The site's monochord and ratios lab follow him: you hear the octave and the fifth, then check that the lengths are 2:1 and 3:2. The Academy's definitions define ratio, and Kepler's third law ties planetary periods to distances by a fixed ratio. For magnitude at rest, see Geometry, Euclid and Trigonometry.

Preparation for philosophy, and for theology

In the tradition the arts are a preparation. Plato calls the mathematical studies the prelude to the strain that must be learned, dialectic (Republic VII, 531d-532b). Nicomachus says that without the four one cannot philosophise properly (I.3); Boethius, that whoever passes them over has lost all philosophy (I.1). Hugh of St Victor explains the names: they are the ways (viae) by which a lively mind enters the secrets of wisdom (Didascalicon III.3). In the medieval scheme philosophy in turn served theology. Bonaventure's On the Reduction of the Arts to Theology distinguishes four lights of knowledge, from mechanical skill and the senses through philosophy to Scripture, and leads the lower three back to theology. Herrad of Landsberg's Hortus deliciarum (12th century, fol. 32r) pictures Philosophia enthroned with the seven arts around her; the manuscript burned in 1870 and is known from copies. That is the tradition's own account; its teaching on habit, proof and ratio can be used whether or not one shares its theology.

Where this could be wrong

The strongest objection is that the order is imposed, not found. Nothing in nature requires seven arts, the substance-and-quantity grounding was laid over an existing list, and modern mathematics does not divide this way. A square's side and diagonal have no common measure (Aristotle, Prior Analytics I.23, 41a26-27; see the proof), so number and magnitude are not interchangeable, and Euclid treats proportion for magnitudes (Book V) apart from proportion for numbers (Book VII). Yet analytic geometry (Descartes, 1637) and the real numbers (Dedekind, 1872) later joined them, so every point on a line can be named by a number. Equal temperament, irrational in every interval but the octave, strains the ratio view of music too.

The best reply is that the scheme was never physics but an order of study: Boethius puts arithmetic first because the other three depend on number (De institutione arithmetica I.1). What would settle it is evidence that this order makes the arts easier to acquire than another; this page knows of none. A Map of Mathematics: Twenty-Five Areas and What Depends on What shows how the modern branches depend on each other, and the curriculum is the place to test the order on yourself.

Try this

1. Take one sentence you wrote this week. Say what it names, what it claims and how it is put to a reader, then fix one problem at each stage.
2. On the monochord, find the octave and the fifth by ear, then check the lengths. Write one sentence on where the ear acted as witness and where reason acted as judge.
3. Explain why grounding the trivium in substance is an interpretation, not Aristotle's division, and name the text you would check.

Further reading

• Aristotle, Categories 4-6; Metaphysics V.13 and VII.1.
• Nicomachus, Introduction to Arithmetic, Book I, trans. M. L. D'Ooge.
• Boethius, Fundamentals of Music, Book I, trans. C. M. Bower.
• Hugh of St Victor, Didascalicon, Book III, trans. J. Taylor.
• Sister Miriam Joseph, The Trivium, chapters 1 to 3.