Mathematics and the Quadrivium

Mathematics and the Quadrivium

Personally, I struggled with Math in school. Faced with a math test, any math test, I froze, cried, banged my head against the desk, and ultimately gave up. I saw mathematics as an isolated “thing” to be conquered. You were either good at math, or you were not.

How little I knew, and how little I was taught, about true mathematics. More than numbers, factorials, and fractions, Mathematics is about relationships – of numbers: how they work with each other, work for us, against us, and can talk about any situation. There are mathematics of money, elections, government, science, music, agriculture, capitalism, socialism, any -ism. Math is language and structure: it is a bridge between all aspects of liberal art. Which leads us to the Bridges of Koenigsberg.

bridgesLeonhard Euler, a Swiss mathematician of the 18th Century solved, sort of, the problem of the Seven Bridges of Koenigsberg (Russia, at the time). Koenigsberg had two islands connected by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point (touching every edge only once). Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The bridges did not meet this condition and therefore, no solution could be found to the problem.

Yet, what this Eulerian circuit eventually did provide is the basis for modern topology , which has expanded into areas of quantum physics, cosmology, biology, computer eulernetworking, and computer programming. For example, the Eulerian cycle or path is used in CMOS circuit design to find an optimal logic gate layouts. For anyone wanting to read the paper outlining these paths in the original Latin, it can be found here.  English translations do exist. A good page on the history of topology is here.

Leonhard Euler was a fascinating individual in that he saw mathematics as something that infused all of life. Though his writings, he made applied mathematics accessible to the layman and his scholastic peers alike. An excellent and thorough biography, written by Walter Gautschi, can be downloaded in PDF form here. With a varied interest in all aspects of mathematics  (arithmetic, geometry, algebra, physics), music, anatomy, physiology, and astronomy, he truly was a man of the “Enlightenment.”  While he was not a Freemason from what I can tell, he seemed to hold much regard for the idea of true science, and creating a better world for his fellow man: a Freemason’s true ideals, to be sure.

The Quadrivium

The Quadrivium

What scholars call the “foundation of Liberal Arts” – the Trivium – is taught in order that one may expand to other subjects, building upon the skills learned. These subjects have been varied over time, based on the philosopher teaching them but they are now generally accepted as mathematics, geometry, music, and astronomy – the Quadrivium. While these subjects were taught by ancient philosophers (Pythagoras, Plato, Aristotle, etc.), they became “the Quadrivium” in the Middle Ages in Western Europe, after Boethius or Cassiodorus had a go at translation.

(Encyclopedia Britannica has an excellent article on Mathematics in the Middle Ages, which discusses the Quadrivium briefly.)

Anicius Manlius Severinus Boethius (usually known simply as Boethius) (c. 480 – 525) was a 6th Century Roman Christian philosopher of the late Roman period. Flavius Magnus Aurelius Cassiodorus Senator (c. 485 – c. 585), commonly known as Cassiodorus, was a Roman statesman and writer, serving in the administration of Theoderic the Great, king of the Ostrogoths.  The former, Boethius, did a great deal to translate most of the ancient philosophers from Greek to Latin. Many of his works on Aristotle were foundational learning in the Middle Ages. Cassiodorus made education his life’s passion, particularly the liberal arts, and worked diligently to ensure classical literature was at the heart of Medieval learning. Both men have been credited with coining the term “Quadrivium,” or “where four roads meet.” Adding to the mix of Medieval education “influencers” is Proclus Lycaeus, one of the last classical philosophers and an ardent translator of Plato. He is considered one of the founding “fathers” of neoplatonism and had a great influence on Medieval education as well. His translations of Plato are peppered with his own ideas of education and philosophy. One of his most interesting books, considered a major work, is “The Platonic Theology.”

sevenLA1For the serious student of the classics, all of these philosophers, in their original Greek or Latin (with English translations alongside the original) can be found in the Loeb Classical Library series. Many used book stores, especially near universities, carry these books and they can be had for about 10$ each. There are hundreds of books but all are quite good as original references (See NOTE below) Back to the Quadrivium…

While many see the Trivium and Quadrivium as “separate,” I think this is a manufacture of our modern educational system. The Trivium are the basics for communicating thought, generating ideas, and conveying those thoughts clearly; yet, like Freemasonry, I don’t know that you would have jumped completely away from your foundations. Plato, in The Republic, does note that the quadrivium subjects, as identified above, should be taught separately. The Pythagorean School divided the subjects up between quantity (mathematics and harmonics, or otherwise known as music) and magnitude (geometry, cosmology or astronomy.) Personally, I find it difficult to talk about music without first having at least fundamental mathematics and exploring both together makes sense. I have not delved into the curriculum of the universities of the Middle Ages in Europe but if someone else has, it would be interesting to hear about it. sevenliberalarts

What I find most fascinating about the art surrounding the Quadrivium (and the Trivium, for that matter) is that nearly all of the plates, pictures, or engravings represent the subject matter as female or feminine. Perhaps it has to do with the receptive qualities of studiousness, or the idea of fecundity or maybe gentleness; whatever the reason, many of the Medieval and Renaissance European depictions show all subjects with a feminine demeanor. Since nearly all scholars in the middle ages in Europe were men, perhaps it was simply a bleed-over of the Medieval ideal of women. I am sure this is another subject for another time.

On an additional side note, I searched for representations of the Quadrivium and Trivium in Islamic art, also knowing full well that Islam is aniconistic. Islam really had begun to gain ground at the last part of the classical period in North Africa & Europe and as such did not really experience the same type of “downfall” or Dark Ages, that Europe did. The schools of Islam continued to develop the subjects of the quadrivium and trivium uninterrupted until Europe “caught up.” In fact, many of the mathematics, geometry, and astronomy texts of the latter Middle Ages were translated from Greek to Syriac Aramaic or from Arabic to Latin, and later taught in Latin universities in Europe.  Suffice to say that Islam did have an impact of the learning of the West, probably much more than most people today are aware.

So, why would the Freemason study the Quadrivium? The answer, to me, is obvious. If the one of the primary studies we must take on is Geometry, we need to understand how number fits into this process. We need Mathematics to understand Geometry, and Music to understand relationship of numbers, working in harmony. Astronomy teaches us our place in universe, and allows us to expand our knowledge of our own earth toward the heavens. Geometry, or the study of the measurement of the earth, is far more than the squares and triangle theorems we all know…and love. It’s about how to apply these numbers to the world around us. As we will see in each of the subjects, they can be taken for their base modern “ideas” or we can expand and overlap them, apply them to the natural world, and thereby become better caretakers of not only the earth we live on but the beings who live on it with us. The idea of a Renaissance Man is one who is well-versed in these foundations and has ideas that expand the world around us. They make the world a better place to live in, now and for the future. The Freemason, to me, embodies this idea completely.

Next stop, the subjects of the Quadrivium. Thank you for joining me!

NOTE For those interested in more of the Loeb Classical Library, but limited access to purchase these books, Harvard University Press has been working to put them online. The link is here:

Individuals can subscribe for a yearly cost, with subsequent years being cheaper, and non-profits can also subscribe for a reduced cost. If you are a serious researcher and you would like primary sources, this library is an excellent resource.