Geometry and the Quadrivium

Geometry and the Quadrivium

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 archimedes_circleso-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.

Additionally, the originally translated letter from Archimedes to Eratosthenes can be downloaded here. (Thank you, JSTOR.)


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.

The History of the All Seeing Eye

The History of the All Seeing Eye

What people in the U.S.A. commonly referred to as the “All Seeing Eye” is more appropriately called the “Eye of Providence.” No, not Rhode Island. This would be the more divine providence – the big guy watching over all of us. Or gal. Or it. Whichever you prefer.

The Eye of Providence is the idea that some divine force is watching over us all, sees our deeds and actions, and judges accordingly. The Eye is associated with the Judgement card of the Tarot: Alchemical texts of the 17th Century and Egypt.

While the Eye of Providence, as noted above, has from the 18th Century C.E. onward been associated with Christianity, the idea of an “all-seeing God” has been around for Millenia. What is called “The Eye of Horus” originally was known as the Wadjet Eye, for the goddess Wadjet.

EyeofHorus

The Eye of Horus

Wadjet, as many can read on the Internet and in several Egyptian Mythology books, was one of the oldest deities in Egypt and dates from the pre-dynastic period. She is associated with Lower Egypt as well as the papyrus. Her name symbolizes the color of blue/green – the color of the papyrus plant. She is nearly always associated with a cobra and is considered the protector of pharaohs and the ruling classes of Egypt.

She’s seen as the Goddess, which represents time, heaven, and hell, and she is an ardent protector, especially of children. Over the millennia, she has been merged with many other goddesses, such as Bast and her sister, Nekhebet, Goddess of Upper Egypt, who is shown as a vulture. Wadjet has also been associated with Buto, the city which first revered her – originally named Per-Wadjet.

According to Herodotus, in his Histories ( Herodotus, The Histories, ii 55, and vii 134), “The Egyptians were also the first to introduce solemn assemblies, processions, and litanies to the gods; of all which the Greeks were taught the use by them. It seems to me a sufficient proof of this that in Egypt these practices have been established from remote antiquity, while in Greece they are only recently known.” There is note of a Temple to Wadjet in Buto (Per-Wadjet) that had an Oracle in it; it was considered that the Greek practice of using Oracles was co-opted from the Egyptians who, as Herodotus states, “taught them in their use.” 

The Eye of Wadget

The Eye of Wadget

Originally, oracles were used to be able to link the individual to the divine. A worshiper would travel to the oracle with a very specific and important question relevant to their lives. Kings would consult oracles to strategize on war or gain insight on divine events which would influence their people, like famine or floods. Using auguries drawn from various animals innards, smoke, visions, the flight of doves, or other “symbols,” the oracle would define the upcoming events based on intuition and divine inspiration. Oracles were eventually replaced by priests and religious figures as a way to connect the individual with their god or goddess, and in a way be a conduit for the divine.

We humans have always been looking for a way to have some kind of communion with the mind of our Divine source, whether it is through other individuals, hallucinogens, runes, tarot, channelling, or simply study. We look for some higher source to tell us what we ought to do, what will help us be successful, happy, healthy, or free. In this case, I find the idea of the Eye of Wadjet to be a symbol that connects us to a dusty, mythic past with some vague idea of what it really meant.

We’ve gone from the protection of a goddess to the protection of a country or our ideals, or maybe even our prosperity. I think this is the most interesting thing in this: the symbols we have in our daily lives are historical treasures that have been modified as our cultures modify them. They come to reflect what, and how we think about the world.

Inspired By Art: Nicholas Roerich’s Symbolic Journey

Inspired By Art: Nicholas Roerich’s Symbolic Journey

The Himalayan paintings of Nicholas Roerich (1874 – 1947), Russian painter and philosopher, mesmerized me when I first laid eyes on the collection. I stared at them for hours. I feel a bit as though these paintings knew me better than anyone. It’s like I understood them, and in turn, they understood me. I became interested in why. 

It could be that as a child my imagination was captured by the breathtaking views of mountains. I was inspired as much by the physical grandeur of their peaks as by the spiritual mysteries harbored within them. Indeed, for me the towering peaks represented the very summit of beauty and spirituality. The tallest mountains I ever climbed were in Zermatt, Switzerland.  I had a truly a remarkable time discovering certain wonders in nature such as glaciers and rich vegetation along with snow-capped points. I would take a rest at several spots which were beautiful and magnificent. There were always more heights to scale, another plateau or peak above, views of the Matterhorn.

Was it this experience that caused me to have a connection with Roerich’s art or something else? Are we drawn to art that mirrors a part of our inner self? What was it specifically about Roerich’s art?

Mount of Five TreasuresIt is virtually impossible to separate the artist’s art from his philosophy. Many contemporaries were skeptical of Roerich’s spiritual mission and tended to dismiss his work as being repetitive and unoriginal. However, those who embraced his philosophy experienced something transformative in those same canvases – never ending spiritual realizations.  A brief survey of his life reveals that Roerich was not an adherent of any one established religion or philosophical movement. His name is universally known not only as Master of the brush but also as a thinker and a builder of life.  His art and writings are an evocation to Beauty, to Knowledge, and to Culture. Art education would pave the way to universal beauty and open the gates to spiritual enlightenment. This vision is nicely captured in his philosophical statement of the Master Institute of United Arts which he formed in New York in 1921:

“Art will unify all humanity.  Art is one – indivisible. Art has its many branches, yet all are one.  Art is the manifestation of the coming synthesis. Art is for all.”

Regardless of whether his lofty ideals will ever be realized, or if you agree with the artist’s art philosophy, his paintings are replete with rich symbolic jewels that are a pure delight for anyone to reflect upon.

19284963678_72350b9c8f_oFor example, Roerich refers often to the great treasures stored deep within the Himalayas. The very name Kanchenjunga, he tells us, means, “the Five Treasures of the Great Snow,” because it contains the five most precious things in the world.  He alludes to them in the painting, Treasure of the Mountain. The setting is deep inside a mountain cave.  Roerich causes us to question what treasures he is referring to? Are they rubies, gold, and diamonds?  Or, upon deeper reflection, is he talking about other more inner treasures?

The gurus hold a lamp. Does it suggest that the Light of Wisdom is the treasure of the mountain?   Roerich reminds us that we must think about what we hold valuable. And – taking this to the deeper level of reflection – we should think very clearly about how we find these symbolic jewels.  What does it take?

Pearl of Searching Flickr mediumIn the Pearl of Searching, another journey is pictured. The teacher-student relationship is a theme in many of Roerich’s paintings. Our eye is drawn to the figures and we identify with them, experiencing the panorama that unfolds behind from their perspective. They have reached a plateau and are celebrating the view and a treasure they have found. What is it? What symbolic meaning do we see?

Perhaps the “Pearl of Great Price” is what gives purpose to life.   The mountains themselves symbolize a spiritual world separate from earth but accessible to those who are attuned to the higher realities. The necklace – a symbol of eternity – signifies that this daily seeking is destined to go on forever, an emblem of immortality. This all causes us to consider, “Why are we here?”

Unquestionably Roerich is one of the most interesting 20th century figures. It seems to me that his life was a symbolic journey documented in his massive body of works, each canvas a masterpiece of daring composition, glowing with color. Himalayan is the word not only for his art, but also a metaphor for his thoughts on the process of transformation.

The path to enlightenment, spiritual ascent, the sacred signs of Satyum, Shivam, Sundaram (Peace, Beauty, Truth) – these are the symbolic themes at the heart of Roerich’s paintings and writings, also at the heart of Freemasonry.  They continually cast a spell on me.  Once we begin on this infinite and exponential journey of discovery, we might stop for a moment to enjoy the view or a treasure found, but there is no turning back. Ultimately, we choose the next plateau we want to rise to.

From Roerich’s “Realm of Light”:

“Let us abandon the past for the future. Let us impel our entire consciousness into the future and let us suffuse it with radiance, for this is within the access of humanity.”

Note: To read further on Nicholas Roerich, check out the Nicholas Roerich Museum.

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.