Quadrivium

When we speak of “classical education”, we speak of a system of learning which provided students with a knowledge of seven liberal arts. These seven arts were divided into two groups. The first three made up the Trivium, and the latter four made up the Quadrivium. Let us consider the ancient Quadrivium.

In the 6th century AD, the Catholic scholar Boethius wrote the following:

“Among all men of ancient authority, it is evident that no one ever ascended to the height of perfection in the disciplines of philosophy, except he whom such was sought by a certain quadrivium.”

Boethius, On Arithmetic

The Quadrivium is that part of the seven liberal arts which consist of the four mathematical arts:  Arithmetic, Geometry, Music and Astronomy.

What are “Mathematics”?

Before we can discuss the Quadrivium, we must first understand what the word “mathematical” means in philosophy.  The best introduction to these “mathematical” arts can be found in the opening chapter of Ptolemy’s Almagest.  Ptolemy writes:

“Aristotle divides theoretical philosophy into three primary categories: Physics, Mathematics and Theology. For everything that exists is composed of matter, form and motion; none of these three can be observed in its substratum by itself, without the others: they can only be imagined.  Now the first cause of the first motion of the universe, if one considers it simply, can be thought of as an invisible and motionless deity; the division of theoretical philosophy concerned with investigating this can be called “Theology“, since this kind of activity, somewhere up in the highest reaches of the universe, can only be imagined, and is completely separated from some perceptible reality. The division of theoretical philosophy which investigates material and ever-moving nature, and which concerns itself with “white”, “hot”, “sweet”, “soft” and suchlike qualities one may call “Physics“; such an order of being is situated (for the most part) amongst corruptible bodies and below the lunar sphere.”

Having discussed the first two divisions of philosophy, Ptolemy now explains the nature of mathematics, which supply the subjects studied in the Quadrivium:

“That division of theoretical philosophy which determines the nature involved in forms and motion from place to place, and which serves to investigate shape, number, size, and place, time and suchlike, one may define as “Mathematics“. Its subject matter falls as it were in the middle between the other two, since, firstly, it can be conceived of both with and without the aid of the senses, and, secondly, it is an attribute of all existing things without exception, both mortal and immortal: for those things which are perpetually changing in their inseparable form, it changes with them, while for eternal things which have an aethereal nature, it keeps their unchanging form unchanged.”

What is the Role of Mathematics in Our Studies?

Ptolemy continues, explaining the excellence of the Quadrivium:

“The first two divisions of theoretical philosophy should rather be called guesswork than knowledge, Theology because of its completely invisible and ungraspable nature, Physics because of the unstable and unclear nature of matter; hence there is no hope that philosophers will ever be agreed about them; and that only Mathematics can provide sure and unshakeable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely Arithmetic and Geometry. Hence we were drawn to the investigation of that part of theoretical philosophy, as far as we were able to the whole of it, but especially to the theory concerning divine and heavenly things. For that alone is devoted to the investigation of the eternally unchanging. For that reason it too can be eternal and unchanging (which is a proper attribute of knowledge) in its own domain, which is neither unclear or disorderly. Furthermore it can work in the domains of the other two divisions of theoretical philosophy no less than they do. For this is the best science to help Theology along its way, since it is the only one which can make a good guess at the nature of that activity which is unmoved and separated; it can do this because it is familiar with the attributes of those beings which are on the one hand perceptible, moving and being moved, but on the other hand eternal and unchanging, I mean the attributes having to do with motions and the arrangements of motions. As for Physics, Mathematics can make a significant contribution. For almost every peculiar attribute of material nature becomes apparent from the peculiarities of its motion from place to place. Thus one can distinguish the corruptible from the incorruptible by whether it undergoes motion in a straight line or in a circle, and heavy from light, and passive from active, by whether it moves towards the centre or away from the centre. With regard to virtuous conduct in practical actions and character, this science, above all things, could make men see clearly; from the constancy, order, symmetry and calm which are associated with the divine, it makes its followers lovers of this divine beauty, accustoming them and reforming their natures, as it were, to a similar spiritual state.

It is this love of the contemplation of the eternal and unchanging which we constantly strive to increase, by studying those parts of these sciences which  have already been mastered by those who approached them in a genuine spirit of enquiry, and by ourselves attempting to contribute as much advancement as has been made possible by the additional time between those people and ourselves.”

With this understood, we can now look into the nature of the Quadrivium.

What is the Quadrivium?

In the opening of his work on the art of Arithmetic, Nicomachus explains the four arts of the Quadrivium.

“If we crave for the goal that is worthy and fitting for man, name, happiness of life–and this is accomplished by philosophy alone, the desire for wisdom–it is reasonable and most necessary to distinguish and systematize he accidental qualities of things.  Things, then, are some of them unified and continuous, which are properly called “magnitudes”.  Others are discontinuous, which are called “multitudes”.  Wisdom, then, must be considered to be the knowledge of these two forms.  Since, however, all multitude and magnitude are infinite, and since sciences are always sciences of limited things, a science dealing with magnitude or multitude per se, could never be formulated.  A science would arise  to deal with something separated from each of them with “quantity” (set off from multitude) and “size” (set off from magnitude).

Nicomachus, Introduction to Arithmetic

I. Arithmetic

Arithmetic is the first of the four arts of the Quadrivium.  Arithmetic is the study of the first division of quantity:  absolute multitude. Through this art we learn the nature of numbers and their different kinds.  The art of Arithmetic is studied in the “Introduction to Arithmetic” by Nicomachus in the Academy’s Classical Arithmetic course.

II. Music

Music is the second of the four arts of the Quadrivium.  Music is the study of the second division of quantity:  relative multitude. Through this art, we learn of the ratios of numbers and their significance, especially as they apply to the practical art of instrumental music. The art of Music is studied in the “Art of Music” by Boethius in the Academy’s  Classical Music course.

III. Geometry

Geometry is the third of the four arts of the Quadrivium.  Geometry is the study of the third division of quantity:  magnitude at rest. Through this art, we learn of the characteristics of different species of solid bodies. The art of Geometry is studied in the “Elements of Geometry” by Euclid in the Academy’s  Classical Geometry course.

IV. Astronomy

Astronomy is the last of the four arts of the Quadrivium.  Astronomy is the study of the fourth division of quantity:  magnitude in motion. Through this art, we learn of the principles of the movements of solid bodies of numbers and their significance, especially as they apply to the study of the heavenly bodies for many practical ends. The art of Astonomy is studied in the “Almagest” by Ptolemy in the Academy’s  Classical Astronomy course.

Here, then, we see that there are four “mathematical arts”, which make up the classical Quadrivium:  Arithmetic, Music, Geometry and Astronomy.

In addition to these true classical Mathematics courses, we also offer students the resources they need to fulfill modern Mathematics requirements in our Modern Arithmetic, Modern Geometry, Algebra and Trigonometry courses.

God bless your studies,
Mr. William C. Michael, Headmaster
Classical Liberal Arts Academy