The Quadrivium: Harmonies of the World

The Numerical Arts

The Trivium and Quadrivium together make the seven liberal arts. We’ve looked briefly at the Trivium of Grammar, Logic, and Rhetoric. Let us take a glance at the Quadrivium.

The four arts of the Quadrivium are Arithmetic, Geometry, Music, and Astronomy. At first this may seem an odd group. We’re more used to seeing something like Algebra, Geometry, Trigonometry, and Calculus. But the Quadrivium does make sense. A man named Boethius (480-524 AD) put it in these terms: Arithmetic is number. Geometry is number in space. Music is number in time. Astronomy is number in space and time. “Well, I understand Arithmetic and Geometry,” someone might say. “But Music and Astronomy? Those are hobbies. How do those latter two prepare a child for a STEM career?” I should explain what math has become and what it used to be.

The Modern Mathematician

If we were to personify the current view of math, we might picture a tall, awkward, and straggly young man. He has slicked hair, a white button down shirt, a pocket protector. His skin is pasty white for want of sun. Under fluorescent lights in a room with no windows at a cold metal desk he calculates. Or rather, a computer calculates. He minds the computer and ponders its calculations. The bridge will fail at a load of eighty tons. The curvature of the plane fuselage is less conducive to flight and more conducive to fiery death. Change the frequency of the radio receiver, unless you want to hear the neighbor’s baby crying over the monitor. The mathematician works with the numbers and finds the solutions. He goes home with some sense of satisfaction in his work – sometimes – but will be very happy if he doesn’t have to think about numbers again until he clocks in the next day. It’s also worth noting that this mathematician does not believe in God. He holds that scientists, using numbers in some fabulous way, have disproven the existence of anything more almighty than the mathematician.

Now this man is a useful man. We want him, or at least we want what he does. But this little caricature identifies four problems with the current state of math. First, in large part math has been removed from the human mind and relegated to computers. Second, math has become limited to the realm of work. Third, math has been brought inside and locked in a sterile room, with the result that we think of numbers as something manufactured by man. And fourth, math has been set in false opposition to a Creator.


Allow me to introduce the four arts of the Quadrivium and paint a different picture of the mathematician. First comes Arithmetic. Arithmetic is the foundation of the Quadrivium, just as Grammar is the foundation of the Trivium. In Arithmetic the student learns to equate numbers with symbols, learns how to count, learns how to add, subtract, multiply, divide. There is much rote memorization: addition tables, multiplication tables, order of operations. It doesn’t take long and Arithmetic can be put to good use. One can count back change, balance a checkbook, calculate cost per ounce and determine which bottle of ketchup gives the better bargain. This is Arithmetic at work.

But Arithmetic also likes to play. “How high can you count?” one of my daughters asked me. “Why, I can count so high,” I replied, “that I could spend the whole rest of my life counting!” Now how did I know that? From experience? No, but because counting is orderly and predictable and follows a set pattern. One could learn to count to a centillion in a fraction of the time it would actually take to do so. And because numbers follow patterns they can be extraordinarily fun.

Consider, for example, the Fibonacci sequence. Can you identify the pattern to these numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144? Each number is the sum of the two previous numbers in the sequence. How many numbers must there be in the sequence in order to pass one million? How can one calculate that without writing out the whole sequence? Do prime numbers come up in the sequence with any predictable regularity? Do odd numbers or even numbers occur more frequently? Why? You see that Arithmetic captivates the mind in a way that can hardly be called work.


On to Geometry! As noted, Geometry takes the numbers from Arithmetic and puts them in space: plotting points in two dimensions on a graph, calculating the surface area or volume of a three-dimensional object, figuring the acreage of a field. Geometry proves itself useful, as did its younger sibling Arithmetic. Say you’re building a garden, 4’ x 8’ x 1’. How many 8’ lengths of 2×6 lumber do you need to construct the walls? You must square the sides as you put them together. How can you use the Pythagorean Theorem (a2 + b2 = c2) to ensure that the corners form right angles? How many cubic feet of soil do you need in order to fill your garden? Or how many gallons of paint do you need to cover the walls of your living room? Or how many gallons of water will it take to fill your swimming pool? Beyond these household applications, Geometry also gives us the architect, the engineer, the packaging specialist, and many other useful vocations.

But why should Geometry limit itself to mere utility when it can also give enjoyment and wonder? Geometry points out the fractal patterns of snowflakes, the order of spots on a leopard’s coat, the scales on fish, the pleasing repetition of the veins and edges of leaves. And try this: Pick up a pineapple. Look at the pattern of the patches on it. Start at the top of the pineapple and trace a curve of patches down and to the right until you get to the bottom. You could go all the way around the pineapple like this and count the number of curves that sweep down and to the right until you arrive back at where you started. And you know what? More often than not the number of curves will be a number from the Fibonacci sequence! You might do it again, tracing the curves down and to the left all the way around the pineapple. The number of those curves is also likely to be a number from the Fibonacci sequence.

You can do the same thing with pinecones, or artichokes. Suddenly the student of Geometry realizes that the Fibonacci sequence isn’t a pattern invented by man as a mathematical curiosity – it is built into the world! Man might be smart enough to notice it, but someone more intelligent made it. Far from being a cold science invented by man in a laboratory, Geometry is an art observed in and learned from the harmonious order of nature.


We continue with Music. Now if an education is merely about learning skills that make money, then many would exclude Music. Very few people make money from it, and those who make the most money from it don’t even understand it. But what if, in addition to the rather simple goal of preparing people for STEM careers, the Quadrivium has a loftier goal? What if the purpose of the Quadrivium is to show the order of the world, to delight mankind with numbers and inspire him to wonder? What if, rather than manufacturing hunched mathematicians and placing them at desks in dim rooms, the Quadrivium purposes to bring its disciples outside, orient their faces toward the heavens, and place on their lips doxologies to the Most High? If that should be the case, then Music and Astronomy are not useless appendages or electives, take them or leave them. Rather, Music and Astronomy are the higher arts that take aim not at money, but at teaching human creatures to love their created-ness.

Music hardly ever goes to work, and when she does she hardly considers it work. She loves to play, and we retain this in our language of Music. We don’t “work” instruments, like scientists in a lab. We play instruments, and in our playing we testify that there is much more to life than money and a job. There’s beauty and delight. But don’t get the wrong idea about Music. She doesn’t play in a chaotic way, like a careless child crashing around the den. She plays in an orderly manner. It’s more fun that way.

Consider a piano. From C to C there are 13 keys, 8 white keys and 5 black keys. The black keys come in sets of 2 or 3, and between the back keys are either 1 or 2 white keys. If you start on C and count up 8 keys (with C as the first key), that’s called a 5th. If you start on C and count 5 keys, that’s a 3rd. If you count 13 keys that’s an octave, from the Latin word octo, 8. Have you noticed the pattern yet? The layout of a piano keyboard can be explained by ratios of numbers from the Fibonacci sequence! These ratios also explain the Circle of Fifths: if you multiply a frequency by 3/2 you get the respective 5th. While much of the Quadrivium shows the beautiful order of the world to the eyes, Music shows that beautiful order to the ears.


Now we come to the highest art of the Quadrivium: Astronomy. Astronomy pulls us from our man-made dwellings and brings us outside into a dwelling made by no man. Astronomy puts a gentle and strong hand under our chins and lifts up our heads. Astronomy turns us from marveling at the work of our hands to marveling at the works of another.

In a way Astronomy makes us uncomfortable. It is supposed to. We tend to think we’re really something. We’re given to puffing ourselves up, whether by worshiping what we’ve made or letting others inflate us with vainglory. Astronomy shows where you really stand in the cosmos. There is a God who perfectly orchestrates the movements of Jupiter and Saturn. What can you do that compares with that?

The Lord questioned Job, “Can you bind the chains of the Pleiades or loose the cords of Orion? Can you lead forth the Mazzaroth in their season, or can you guide the Bear with its children? Do you know the ordinances of the heavens? Can you establish their rule on earth?” (Job 38:31-33). Astronomy asks you the same questions. And we must answer with Job, “Behold, I am of small account” (Job 40:4).

The greatest astronomers have been Christians, perhaps because they’re not afraid to face the grandeur of God or their own smallness. They know from the Scriptures that while we are of no account by ourselves, we have been made in the image of God and the Son of God has shown us such favor that he died for us. The Christian, therefore, does not have to hide from Almighty God, but is free to marvel at the work of his hands.

This marveling is awfully fun, too. The astronomer Johannes Kepler (1571-1630 AD) is a good example. In his book Harmonies of the World he shows how geometric shapes inscribed in one another come close to the spatial arrangement of planetary orbits. He observes that one can represent the paths of the heavenly bodies with numerical ratios (the fact that the ratio 3/2 factors heavily in this really shouldn’t surprise us by this point).

Kepler also takes the ratios of the planets and converts them to musical ratios. He then uses musical notation to write out the harmonies that the planets “sing”: Saturn and Jupiter sing bass, Mars sings tenor, Earth and Venus sing alto, Mercury sings soprano. He portrays the heavens as a choir singing four-part harmony in praise of their Creator.

In doing this Kepler is not projecting man’s music into the heavens. It’s rather the opposite. Kepler writes:

Accordingly, you won’t wonder any more that a very excellent order of sounds or pitches in a musical system or scale has been set up by men, since you see that they are doing nothing else in this business except to play the apes [i.e. imitators] of God the Creator and to act out, as it were, a certain drama of the ordination of the celestial movements (Harmonies of the World, Chapter 5).

The Classical Mathematician

Having covered the four arts of the Quadrivium, let us conclude with this question: what kind of mathematician does the Quadrivium make? First of all, the Quadrivium makes all of its students mathematicians, and for no other reason than the pure joy of numbers. If some go on to make money with these numbers, fine, but the Quadrivium is not concerned about forming money-makers.

The Quadrivium forms a person who is so conversant in the language of numbers that he can figure even complex equations without the aid of a calculator. The Quadrivium forms a person who finds so much wonder and fun in the numbers and ratios and harmonies of the world that he can’t think of it as work. The Quadrivium forms a person who understands that numbers are received from the created world, and are not manufactured by men in their closets. Most importantly the Quadrivium forms a person who understands the whole universe in mathematical terms, and rather than misusing his knowledge to explain God away, he breaks forth in high doxology.

The mathematician formed by the Quadrivium may still be tall, awkward, and straggly. He may still have slicked hair, a white button down shirt, and a pocket protector. He certainly still uses his knowledge of numbers in service to others. But his sleeves are rolled up, his skin knows the light of the sun, his face inclines toward the heavens, and he praises God the Father Almighty, Maker of heaven and earth:

Great is our Lord and great His virtue and of His wisdom there is no number: praise Him, ye heavens, praise Him, ye sun, moon, and planets, use every sense for perceiving, every tongue for declaring your Creator. Praise Him, ye celestial harmonies, praise Him, ye judges of the harmonies uncovered…: and thou my soul, praise the Lord thy Creator, as long as I shall be: for out of Him and through Him and in Him are all things, both the sensible and the intelligible; for both those whereof we are utterly ignorant and those which we know are the least part of them; because there is still more beyond. To Him be praise, honour, and glory, world without end. Amen. (Harmonies of the World, Chapter 10).

Harmonies of the World by Johannes Kepler, 1619
Translated by Charles Glenn Wallis, 1939

Woodcut: The Seventh Day of Creation by Julius Schnorr von Carolsfeld, 1794-1872
Image provided courtesy of the Pitts Theology Library, Candler School of Theology, Emory University.

The Trivium: The Architecture of Language

The Seven Liberal Arts

The framework of a classical education is the liberal arts. The word “liberal” is from the Latin word liberalis, relating to a free person. At another time we’ll look at what it meant to be a free person in the classical world, and what a liberal arts education frees one from. At this time we’ll look at what the liberal arts are. There are seven liberal arts: Grammar, Logic, Rhetoric, Arithmetic, Geometry, Music, and Astronomy. The seven liberal arts are distinguished by two subcategories: the trivium (tri – three) and the quadrivium (quad – four). To explain the distinction in the simplest way possible we might say that the trivium focuses on letters and the quadrivium focuses on numbers. In this article I’ll explain the trivium.

The Trivium: Grammar

We can think of the trivium in terms of architecture. First the architect must understand his building materials and tools. In language this involves learning the alphabet, learning how to combine letters into phonenes and phonemes into words and words into sentences. Vocabulary is the lumberyard and quarry. Syntax is the mortar, the nails. The first art of the trivium is Grammar, which teaches one to select the proper pieces of wood, pick up a hammer, and stick a few boards together in a useful and orderly fashion.

Learning grammar inevitably includes whacking thumbs and dropping bricks and prying out bent nails. There are important distinctions to make, and mastering them takes time and frustration and fortitude. Pronouns have different declensions depending on their role in a sentence: I, he, she for subject; my, his her for possessive; me, him, her for direct object. Verbs sometimes conjugate predictably: walk, walking, walked. Sometime verbs conjugate quite unpredictably: be, were/was, been, am, is, are; go, goes, went, going, gone. There are transitive verbs and intransitive verbs, and many of us still bruise our fingers with lay (transitive) and lie (intransitive) – “he lays the baby in the crib and then lies in bed.” There’s punctuation and spelling and learning to count syllables. When is it “less” and when is it “fewer”? When “your” and when “you’re”?

The teacher mercilessly drills the basics. Each question has its set answer that is not up for discussion or debate. It is not that the teacher devalues self-expression. The teacher simply recognizes that without basic grammar the students cannot adequately express themselves. So in love for the students and desiring to impart a great gift the teacher says, “Shut your mouths and open your ears. This is how it is.” Drill, drill, drill, drill, drill.

This is exhausting work, but when the junior architect builds his first little wooden bridge or stone wall, when he spells every word in a sentence correctly and properly aligns persons and numbers and tenses of the nouns and verbs, when the bricks are level and the studs equally spaced – well! there is great satisfaction, and a healthy pride, and hope for this great undertaking of language.


The second art of the trivium is Logic, also called Dialectic. This art has to do with reason and argumentation. In Logic the student learns to make sound arguments and analyze the arguments of others. A key tool for this process is the syllogism, which consists of a major premise, a minor premise, and a conclusion. To give an oft-used example: All men are mortal. Socrates is a man. Therefore Socrates is mortal. That syllogism is both valid and sound: valid because the conclusion flows logically from the premises, and sound because, in addition to being valid, the premises are also true.

I could say: All cats can fly. I have a cat. Therefore my cat can fly. It is a valid argument, but neither of the premises are true, and therefore the argument is not sound. It would be like building a staircase out of balsa wood. Yes, it’s a staircase. But it’s going to collapse under me if I try to climb it. Or I could say: All people who eat at Olive Garden are happy (see how they smile in this commercial?). You want to be happy. You should eat at Olive Garden or you will never be happy. The logic behind a syllogism like this is the equivalent of cutting a hole in the upper story of a house and calling it a staircase. More frequently it goes by the name “advertising.”

In Logic the junior architect learns how to join walls and dovetail beams, to pattern trusses and support ceilings. While in Grammar he learned to make window frames and lay floorboards, in Logic he combines these elements to make rooms, and creates passages between the rooms and the various floors. The solid structure of a house results. The teacher drills the forms of syllogisms and gives many examples of good and bad arguments. When the students are not reciting forms and logical fallacies they’re debating.

And they’re not stuck in childish debates, like whether pepperoni or pineapple makes a better pizza topping. No, they can go beyond mere subjective preferences. They’ve been instructed in Grammar and Logic, and moreover have studied these arts through good literature that has instilled good moral virtue. They can argue about right and wrong, better and worse, without having to rely on their personal opinions. Never will they say, “That might be true for you, but it’s not for me.” They have become acquainted with the real world, the world of absolutes, the world where a statement can be objectively true or false. They have learned to navigate this world and speak its language. The strength of their architecture shows it.


The third art of the trivium is Rhetoric, perhaps most simply defined as the art of persuasion. In tandem with sound arguments, Rhetoric moves an audience toward what is good and true. Bare Logic might say, “Here’s a bedroom, and bathroom, a kitchen, a living room, and a laundry room. You have everything you need. Move in.” Logic isn’t wrong in thinking that people should believe its arguments on the basis of soundness and truth. Logic can be something of a pragmatist if we leave it alone. “You need a table? Ok, here’s a barrel with a scrap of plywood on it. A table.”

But Rhetoric says, “No, friend Logic, not like that. I agree with your design. A flat surface on a support, that’s good. That’s sound. But use this mahogany for the table top. Let’s cut it into a circle, bevel the edges, stain it, and varnish it. And let’s use a thinner support. I’ll spin something on the lathe. Ah, yes, see how it complements the top! It’s much more pleasing, and now you don’t bash your knees on the support when you sit down.”

That’s what Rhetoric does. It weaves thoughts and arguments into a cohesive whole, it removes unnecessary stumbling blocks, it beautifies Logic and pleases the audience. Rhetoric runs through the house and puts arches over doors and exotic wood on the floors. It dresses stones for the exterior walls, landscapes the yard, builds columns with Corinthian capitals for the porch. And when the junior architect has been trained in Rhetoric, he’s no longer a junior architect. He has what he needs to be a master. The student can choose the one right word out of a million and put it in service of sound logic and adorn truth with the beauty that truth deserves. The student can analyze, respond, express, and convince. When lies come calling in their pretty disguises he can spot them instantly, strip back their sequined cloaks, and show them for what they are. And meanwhile, from his mouth and his pen the student sends goodness and truth and beauty into the world.

Restore the Trivium

This sort of student, this sort of architect, has become a rare creature. Many who have some small command of rhetoric put it in service of invalid and unsound arguments. Many more don’t understand enough logic to tell truth from lie. What happened? Two main things. First, in many schools objective truth was told, “It’s a nice day outside. Why don’t you take a walk and, uh, don’t come back.” The history and consequences of this deserve several articles, and we needn’t delve into these details as we near the end of this one.

The second reason why logicians and rhetoricians are rare is that in many schools self-expression has become the starting point of education and not one of its ends. “After all,” educators argue, “who wants to diagram sentences and learn about glottal stops? Here’s a piece of paper and a pencil. Write a story. Tell me about your favorite color.” This is like giving a man a clarinet and saying, “Play this instrument. Express yourself.” All that follows is squeaking from the clarinet, frustration from the man, and a desire for earplugs from everyone who must listen to him. But because this has been going on for so long, we don’t even desire the earplugs anymore. “Hey, not bad,” we say. “You sound like the rest of the clarinet players.”

But where has this left us? We can’t engage in an argument without taking it personally. We lack the logic necessary to make a case or critique one. Civil discourse seems to exist in name only. And it’s not that persuasion has stopped. It’s that persuasion has fallen to the level of base manipulation. Without good logic or rhetoric we are forsaken to the winds, blustered about by slogans and soundbites. We’re subject to pithy statements whose conclusions have no premises. These slogans tickle the ears and achieve some emotional effect. Men with wrecking balls point at a cardboard box and say, “Look at this mansion! It has everything you need. It’s a lovely place. You should move in.” And to their own harm many do.

Our children deserve better. We owe them better. So let’s give them the trivium. Let’s teach them to hew stones and plane boards. Let’s teach them to build houses and town halls and cathedrals. Let’s make them reasonable and eloquent in matters of home and state and Church. And while we’re at it, let’s abandon our own shabby boxes and aspire to something better.

Painting: “St. Paul Preaching in Athens” by Raphael, 1515