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Quadrivium


Pythagoras (570-495 BC)

When we think of Mathematics today, we think of modern, practical mathematics, which begin with basic calculations and lead up to Trigonometry and, maybe, Calculus.  This has almost nothing to do with classical Mathematics.  Moving from a new math curriculum to a Mennonite textbook doesn’t make your modern math “classical”.  Saxon Math is not “classical” Mathematics.  Modern college prep math courses, no matter how they are repackaged, cannot become “classical” Mathematics.  Come on, Christians.  We have to know better than this.

We are creatures made in the image of God, created to know, love and serve God and to be happy with him in eternity.  Our nature and purpose transcends all created things.  St. Paul taught us, “If you have been raised with Christ”–and all Christians have, in Baptism–“then seek the things which are above, where Christ is seated at the right hand of God.”  Christians are not to be seeking the things that unbelievers seek, but higher and greater things–eternal and unchanging things.  These are to be to the objects of Christian studies.  These are the sources of that peace and order which is to mark the minds and lives of Christian people in this world.

The ancients understood these things, which means that if they studied Mathematics, it was not for business and college admission (obviously), but for this end.  Socrates asked,“What sort of knowledge is there which would draw the soul from temporal things to eternal things?”  The answer to that question, found in Plato’s Republic, is Mathematics.  He says, “this is the easiest way for the soul to rise from changing things to truth and unchanging things.”

The Quadrivium

Of Mathematics, the Catholic philosopher Boethius (480-524 AD) explained the following:

“Among the ancients it was agreed that no one ascended to the summit of perfection in the disciplines of philosophy unless it was sought by a certain, as it were, quadrivium.”

Boethius introduced the term “quadrivium” to describe the four mathematical arts in the opening lines of his work “De Arithmetica

The “four ways” that Boethius refers to are explained by the mathematician Nicomachus (60-120 AD).   There are two division of “essences”, or unchanging things that Socrates spoke of above:  multitudes and magnitudes.  Multitudes are collections of separate objects, such as a flock of sheep or a dozen eggs.  Magnitudes are continuous things, such as one sheep or one egg.

Multitudes are, themselves, of two kinds.  One kind, Absolute multitude,  is that which is viewed by itself and so named, such as “even”, “odd”, “perfect” and so on.  The other, Relative Multitude, is that which is viewed in relation to some other thing and is so named, such as “half”, “double”, “triple”, and so on.  Magnitudes are, likewise, of two kinds.  One kind is Magnitude at rest, the other is Magnitude in motion.

As there are four kinds of essences, there must be four different sciences by which each of these are investigated.  These four arts, as listed below, make up the Quadrivium:

  •  Arithmetic:  the doctrine of absolute multitude
  •  Music:  the doctrine of relative multitude
  •  Geometry:  the doctrine of magnitude at rest
  •  Astronomy:  the doctrine of magnitude in motion

Now, while we may see the names of these four studies used in schools today, we can be certain that they do not have the meanings that they had among the philosophers.  This is one of the greatest problems:  the confusion caused by the continued use of names that no longer mean what they did in the past.  Nicomachus warns us that:

“Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is Wisdom.”

Courses Available


The CLAA, in keeping with the teaching of the ancients, offers courses in each of the four arts of the Quadrivium.  We add “Classical” to each of the names to distinguish them from modern subjects that have maintained their names while abandoning the ancient principles:

If you have any questions about our courses in the classical mathematical arts, please contact us

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