Triangles Everywhere

I was recently exploring the idea of the triangle – its form, function, stability, and meanings. In Freemasonry, as in many traditions, the triangle holds significant influence in symbolic meanings.

A triangle is a polygon with three sides and three vertices. There are many forms of triangles – right, equilateral, obtuse, acute, isosceles, and scalene. There are also oblique and degenerate triangles. Triangles may be multiple types. Triangles are generally believed to be two-dimensional objects whose interior angles, at least in Euclidean space, equal 180 degrees. They can be various shapes but the ones most often seen are right triangles and equilateral triangles.

Of the triangle knowledge from history, the famous philosophers Pythagoras, Plato, and Euclid are known best for theorems, ideas, and esoteric supposition surrounding the form. The form is so basic that it’s most likely older than written history. Ancient petroglyphs, such as those from Columbia, the Sierras in North America, and Mexico, show humans with bodies and heads in the form of triangles. This is a basic shape that mimicked the human form, with wide shoulders and narrow waist, or a wide head crown and narrow chin. There isn’t anything to indicate, in-depth, the symbolic meaning of the triangle other than it being incorporated into the human form.

The Egyptians used the triangle quite often, generally in the realignment of land plots after the Nile floods but also in architecture. In a 2000 thesis article regarding the “sacred triangle,” the author asserts that Egyptians knew and used, even in the Old Kingdom, the “sacred triangle” of 3:4:5. Indeed, the author goes on to state that using straight vertices, or a “simple, straight vertical pole,” to find location or identify specific time of day or days of the year. While this is a heavy-mathematics article, the reader might find some deeper, symbolic meanings in the geometry.

During the 6th C. B.C.E., the School of Pythagoras became known for its theorem regarding the formation of the ‘sacred triangle.’ Pythagoras left no mathematical writings of his own, while Euclid and Plato did. Thales of Miletus is really the creator of basic mathematics and geometry, and probably the first to give us theorems about the triangle. Pythagoras, who created the words philosophy and mathematics, is more well-known and did much to bring the form of the triangle into deeper meaning.

To Pythagoras, the number 10 was the holiest of numbers; the tetractys is a triangle form of 10 dots, created by interlinking the dots into nine triangles forming the 10th, larger triangle. It is used to symbolize the creative forces of the universe. From ancient-symbol.com, “In the figure, the first row has a single point that is representative of the Creator, the active principle, the divine power behind all creation and is associated with wisdom. The second row contains two points that represent the passive principle and are associated with friction, movement, impulse, strength, and courage. The third row with three points signifies the world coming out of the union of the above two, a union of physical and mental balance and is associated with harmony. The fourth row has four points that represent the four liberal arts & sciences that complete the world. These four points symbolize the four elements of earth, fire, air, and water.” This was, generally speaking, the first time that the philosophical meaning of a number, its holiness and perfection, being derived from pure mathematical reasoning rather than from inductive reasoning. It was more than the total of our fingers on our hands. Another interesting article on the triangle and tetractys, among other things, can be found here: http://www.projectawe.org/blog/2015/12/21/up-and-down-the-monochord-part-iii-triangle-trinity-unity. The author of this blog does a very good and thorough job of digging into these ideas, and I would highly encourage everyone interested in these subjects to read it.

In the alchemical writings of the Middle Ages, the classical elements of hermeticism were based off the form of the triangle, turned upward or down, with a line to denote the opposite or without to indicate the base elements. The conjoining of fire and water is indicative of balance and achieving perfection. The triangle is also seen in the “triangle of art” also known as Solomon’s Triangle. The circle in that triangle represents the space where spirits are called, with the triangle representative of the safe space from which the magician worked.

Triangles in astrology are seen as very positive, and a grand trine, or golden triangle, is seen as a creative, harmonious flow of energy in a person’s life; they generally are composed of the objects being in the same elements, in the form of an equilateral triangle.

Triangles are a form of stability, where two extremes are balanced by a third point. Triangles are everywhere in Freemasonry, overt and subtle, and have different stories surrounding each. These different stories speak to individuals differently even if the core remains the same; depending on the degree being worked and studied, the aspirant may find different aspects of the same truth. These truths are not much different than the ancient Egyptians and Greeks found and used in their daily lives. There are always extremes and balance is achieved by that third, divine point. One might also see that all emanates from the Divine, the single point, which may also turn into that point within a circle which is perfect balance. The perfect man may be the one who finds equilibrium during whatever storm shakes him. Taking this symbolism into our daily lives and applying it to our relationships with people is really the value of the study of symbol. We can work toward being the middle point between extremes, able to see both sides in equal measure. A more holistic view of those things that permeate our lives creates a better person.

Whist sitting in school, slaving away with compasses and a ruler, one hardly remembers that geometry is the study of the measurement of the earth. Earth. The thing we sit on, utilize, and finally rest in when this is all over. The geometry in schools today looks nothing like the geometry of 3000 years ago. It is difficult to divorce geometry from the other liberal arts when we take into consideration the scale to while discoveries are interconnected. Geometry arose from the needs of agriculture, civilization, and war. For so much of this, we can thank Archimedes of Syracuse. A student of Euclid in the 3rd c. BCE, his advances in the field of geometry furthered irrigation (Archimedes’ Screw), astronomy (the first planetarium), and weights & measures (Archimedes’ Principle). The most interesting, to me, is The Method of Exhaustion (remember Dialectica) also known as “The Method” or “Archimedes’ Method.”

“…, to estimate the area of a circle, he constructed a larger polygon outside the circle and a smaller one inside it. He first enclosed the circle in a triangle, then in a square, pentagon, hexagon, etc, etc, each time approximating the area of the circle more closely. By this so-called ‘method of exhaustion’ (or simply ‘Archimedes’ Method’), he effectively homed in on a value for one of the most important numbers in all of mathematics, π.” 1

Linked together with this Method is the “Method of Mechanical Theorems.” Proofs are everything to the mathematician, and in his Method of Mechanical Theorems, Archimedes had none that would be accepted. He set out using Eudoxus’ The Method of Exhaustion to prove what he knew to be true. In a letter to Eratosthenes, in manuscripts discovered in 1906, Archimedes outlines his thought processes. This document is known as the Archimedes Palimpsest.

Certain theorems first became clear to me by means of a mechanical method. Then, however, they had to be proved geometrically since the method provided no real proof. It is obviously easier to find a proof when we have already learned something about the question by means of the method than it is to find one without such advance knowledge.

The importance of these discoveries and the methods by which Archimedes came to them may be obvious – who doesn’t need π? However, it is also fascinating to peer inside the mathematician’s mind and view it with a Freemason’s perspective. Here was a man who could see the Plan, understand the Plan, and only needed to bring it to life: a divine spark of wisdom, the will to discover, and beauty in its presentation.

For an interesting and short expose on The Method and the “Archimedes Palmipsest,” whence this Method is documented, review  “The Illustrated Method of Archimedes” by  Andre Koch Torres Assis and Ceno Pietro Magnaghi. The PDF can be found here.

Just a note (1): The Story of Mathematics, Luke Mastin – http://www.storyofmathematics.com/hellenistic_archimedes.html – I’ve done my best to verify statements here, and so should you.

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.”

For 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.

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: http://www.hup.harvard.edu/features/loeb/digital.html.

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.

The Seven Liberal Arts – The Trivium

There is a real affinity for the goals of Freemasonry and the Seven Liberal Arts. From earliest teachings, we see that they are the foundation of many degree rites, the first of which is the FellowCraft Degree. To understand why this is, I think we must first understand the structure of the Seven Liberal Arts and what their history is.

The Liberal Arts have been, from antiquity, been the foundation stone upon which knowledge of the natural world rests. The seven liberal arts have been utilized since ancient Greece. Plato and Pythagoras were first in codifying their importance; the flowering of our western understanding of the liberal arts took place in medieval education systems, where they were categorized into the Trivium and the Quadrivium. Grammar, Logic, Rhetoric are the Trivium, and Arithmetic, Geometry, Music, and Astronomy are the Quadrivium. The Trivium combines the use of the senses with knowledge to lay the foundation for further study. The Quadrivium was considered to be the higher level education for the philosopher, and employed the use of the Trivium to be able to compose higher ideas and thereby, expand the knowledge of the human condition.

Freemasons the world over have expounded on the Seven Liberal Arts ad infinitum. All you need to do is search Freemasonry and Seven Liberal Arts, and you get a great deal of regurgitated drivel. That is not what I am striving to do in this next series. Here, my goal is to simply explain why the Seven Liberal Arts seem to have a kinship with Freemasonry, and perhaps provide small examples of each – with and without a Freemasonic connection. It’s up to you, the reader, to decide what you’d like to do with the information.

Plato’s Dialogues explain the curriculum outlined in detail and for any serious student of liberal arts, Plato is required reading. I, therefore, will not relate these concepts here. Suffice to say that the study of the Liberal Arts is more of a study of knowledge than it is of any specific actual data and information. As we may have learned by now, knowledge without application is dead and useless. Knowledge in the pursuit of higher ideals and higher ideas is more valuable than… than… well, you get the idea. Remember, one of the goals of Freemasonry is to better the human condition while standing up in defiance of falsehood, ignorance, and hatred. How do we do that if we are not searching to better our communication and knowledge, and the ways to bring both to life?

The Trivium is, as I said above, the foundation stone of the Seven Liberal Arts and really provides us the method and ability to communicate. It is composed of Grammar, Logic, and Rhetoric.

• Grammar: Knowledge and Learning of Language
• Logic: Reasoning, Questioning, and Thinking with Language
• Rhetoric: Directing, moving, and Persuading using Language

While these all seem to be in relation to language, they are much more than language. They are the skills involved in achieving these ends. Therefore, the study of Grammar is also the study of history, geography, reading, and writing. It is basic, absolutely, but more encompassing than simply learning one’s ABCs and how to put pen on paper and write. Logic is about how we learn – we use our senses to experience, put our minds to thought, question, and experiment. We learn to ask the correct questions to achieve the answers we seek. They are not provided to us – we must seek them out and test for ourselves. Finally, rhetoric is the ability to take what we have learned with grammar and dialectic and put them firmly into the hands of an audience we are attempting to persuade. Rhetoric uses emotional discourse, thoughtfully created and properly applied, to communicate new ideas.

If it is not clear to the Freemason now why at least the Trivium is not important, one might want to question what they have actually learned while being a Freemason. Many may think that Freemasonry is all about enlightenment, walking in squares, or religious meanings. It might be those things to some but I think the true goals of Freemasonry are to provide a framework of how to be in the world, to make that world better for those that follow us but more importantly, for our own betterment. We cannot communicate lofty ideals via ritual alone – we need to be able to express what we have learned to a wider audience, to bring new thoughts to a wider world. To me, when we talk about service to the world, there is no greater service than being a hand-up to the betterment of the human condition and we do that by “teaching a man how to fish.” Study of the Liberal Arts is by one means to catch that “fish.”