Return to Free Library Return to Number Menu Previous Article Next Article “A study of Sacred Geometry requires an immersion into the history and meaning of the archaic cultures for whom it provided a vehicle to produce some of the most awe-inspiring demonstrations of symbolic and sacred architecture to have been conceived and executed by the mind and hand of mankind, while at the same time providing a path to a deepened spiritual awareness of the fundamental principles of creation.” Randall Carlson The exact origin of the study of sacred geometry is unknown, yet we can be sure it is quite ancient. One aspect of ancient study is the Seven Liberal Arts. The Seven Liberal Arts consist of the Trivium and the Quadrivium. The Trivium, meaning, ‘the place where the three roads meet’ stretches back to the 8th century and the Carolingian Renaissance. The Quadrivium is far older. Although the term ‘Quadrivium’, meaning ‘four ways’, wasn’t coined until Boethius in the 6th century, the study of the Quadrivium goes back to at least the time of Pythagoras. However traces of it can be found in Babylonian, Chaldean, ancient Egyptian and ancient Hindu cultures. The Trivium and Quadrivium together prepare the student for the serious study of philosophy.

The Trivium The Trivium consists of Grammar, Logic and Rhetoric. Grammar refers to the mechanics of language and the defining of objects and information perceived by the five senses (Input or facts, data, terms, basic skills). Logic refers to the mechanics of thought and analysis. This includes identifying fallacious arguments and statements, removing contradictions, and producing factual knowledge that can be trusted. (Processing, or What and why?) Rhetoric refers to the application of language in order to understand and persuade the listener and reader. (The proper use of knowledge and understanding.) The Trivium “is the knowledge (grammar) now understood (logic) being transmitted outwards as wisdom (rhetoric).”1 The three together represent “Mastery” or true learning. Below we will see that the Trivium has yet another meaning, that of the True, the Good, and the Beautiful. These are the properties of being that correspond to human reality. That is, science corresponds to truth; the arts to beauty; and spirituality to goodness.

The Quadrivium The Quadrivium consists of Arithmetic, Geometry, Harmony and Cosmology. Arithmetic is Number. It deals with quantities. It is ‘Pure Number’. Geometry is Number in Space. It deals with magnitude at rest. It is ‘Stationary Number’. Harmony/Music is Number in Time. It deals with relations between quantities (ratios and proportions). It is ‘Applied Number’. Astronomy is Number in Space & Time. It deals with magnitude inherently moving. It is ‘Moving Number’. It can be seen that ‘Number’ itself – that is, ‘Pure Number’ is the foundation for all the other branches of study. Plato said, “Numbers are the highest degree of knowledge, it is knowledge itself.” He insisted number to be the first study to embark upon as it “frees the mind from attending to actual objects and raises it to the realm of abstract principles and archetypes.”2 Pythagoras said, “All is number.” St. Augustine said, “Numbers are the thoughts of God.” The Quadrivium was the major intellectual discipline of classical education consisting of Number, Geometry, Harmony, and Cosmology. A complete study of these topics were formulated and taught by Pythagoras as the ‘Tetraktys’ around 500 BCE. It was also taught by Plato and outlined in The Republic (380 BCE). It was next taught by Proclus (412-485 AD). Robert Lawlor explains, “The implicit goal of this education was to enable the mind to become a channel through which the ‘earth’ (the level of manifested form) could receive the abstract, cosmic life of the heavens.” Keith Critchlow adds, “All these studies offer a safe and reliable ladder to reach the simultaneous values of the True, the Good, and the Beautiful [the Trivium]…The goal of studying these subjects was to climb back (up) to Unity through simplification based on the understanding gained by engaging in each area of the Quadrivium. The goal lay in finding their source (traditionally this was the sole purpose of the search for knowledge)…Thus, time and wisdom tested, the Quadrivium offers the sincere seeker the opportunity to regain their own inner understanding of the integral nature of the universe, with themselves as an inseparable part.”

Pythagoras Before we explore each of the four studies of the Quadrivium in more detail, we will take a closer look at the life and teachings of Pythagoras, as he had one of the most profound impacts on the consciousness of western philosophy and science. Pythagoras was born around 570 BCE and died around 495 BCE. Manly P. Hall tells us in the Secret Teachings of All Ages that strange legends surround his birth. His life parallels the life of Jesus in many fascinating ways. Pythagoras was believed to have come from an immaculate conception like Jesus. They were both natives of Mediterranean countries. They both had fathers who were prophetically informed that their wife would bring forth a son who would be a benefactor of mankind. They were both born when their mother was on a journey. Both of their mothers claimed to have had contact with a ‘holy ghost’ and both were known as the ‘son of God.’ Pythagoras was thoroughly knowledgeable in Oriental and Occidental esotericism, as was Jesus. He was instructed by Rabbis in the secret traditions of Moses and he was initiated into the following Mysteries: Egyptian, Babylonian, Chaldean, Isis (in Egypt), Adonis (in Syria), and the Brahmans (in Hindustan). Mark Booth, in The Secret History of the World, has this to say about Pythagoras: “Pythagoras was born on the prosperous Greek island of Samos in about 575 BC, as the first blocks of marble were being placed one on top of the other on the Acropolis in Athens. No individual has had a greater influence on the evolution of Western esoteric thought. Pythagoras was regarded as a demi-god during his lifetime. Like Jesus Christ, nothing he wrote has come down to us, only a few collected sayings and commentaries and stories written by disciples. His wisdom was the result of years of research and multiple initiations into Mystery schools. He spent twenty-two years learning the secrets of the Egyptian initiate priests. He also studied with the Magi in Babylon and the descendants of the Rishis in India, where a memory was preserved of the great wonder-worker they called Yaivancharya. Pythagoras was seeking to synthesize esoteric thought from all around the world into a comprehensive cosmo-conception – what Leibniz, the seventeenth-century mathematician and Cabalist, would later call the Perennial Philosophy.” Manly P Hall goes on to tell us that Pythagoras coined the word philosopher– meaning ‘one who is attempting to find out’. He demanded of all who came to him for study a familiarity with arithmetic, music, astronomy, and geometry. He conceived mathematics to be the most sacred and exact of all the sciences. He established a community where all the members were of mutual assistance to one another in the common attainment of the higher sciences. He introduced the discipline of retrospection as essential to the development of the spiritual mind. He taught nothing to his disciples before the discipline of silence – silence being the first rudiment of contemplation. The God of Pythagoras was the Monad, or the ‘One that is Everything.’ Manly P Hall explains: to Pythagoras God is “the Supreme Mind distributed throughout all parts of the Universe – the Cause of all things, the Intelligence of all things, and the Power within all things.” The ‘Motion of God’ was circular. The ‘Body of God’ was composed of light; and the ‘Nature of God’ was truth. Pythagoras taught reincarnation. He also taught moderation in all things rather than excess in anything. He warned his disciples when they pray they should not pray for themselves. He declared eating meat clouded the reasoning faculties, and he discovered music had great therapeutic power. He was opposed to surgery in all its forms and would not permit the disfigurement of the body. He also believed all planets were alive and suns and planets were magnificent deities worthy of adoration and respect. Pythagoras had many aphorisms, including: “Above all things, respect yourself.” “Do not yield to temptation except when you agree to be untrue to yourself.” “All bonds without friendship are shackles, and there is no virtue in their maintenance.” “Wisdom is the understanding of the source or cause of all things.” In The Golden Verses of the Pythagoreans Pythagoras said, “And thou shalt know that Law hath established the inner nature of all things alike.” The ‘Pythagorean Y’ represented the ‘Forking of the Ways’. It symbolized the power of choice and the sanctity of free will. This choice refers to choosing to work for the ‘Greatest Good’ (Unity) or choosing to work for the ‘Individual Selfish Good’ (Separation). One cannot choose both. The ‘Pythagorean Y’ represents one of the core precepts of the esoteric stream of knowledge, one that shows up in literature, art, and many, many spiritual traditions. It represents the necessity for each person to consciously make a choice between the two paths – the path of working for the greatest good, or the path of working for the selfish good. Until one makes the choice, they will be swept from shore to shore, battered among the rocks in the chasm of indifference. Life without a dedicated direction leads to chaos, confusion and pain. Once a choice is made and the person honestly attempts to live accordingly, the consequences will be reaped, whether they be positive consequences or negative consequences. It all depends on the choice. Of course it is wise to remember, as it is said in Led Zeppelin’s Stairway to Heaven, “Yes, there are two paths you can go by, but in the long run there’s still time to change the road you’re on.” “The neophyte must then choose whether he will take the left-hand path and, following the dictates of his lower nature, enter upon a span of folly and thoughtlessness which will inevitably result in his undoing, or whether he will take the right-hand road and through integrity, industry, and sincerity ultimately regain union with the immortals in the superior spheres.” Manly P Hall Pythagoras also developed a complex and beautiful system of mathematics, geometry, music and color. He said, “All things consist of three,” and “Establish the triangle and the problem is two-thirds solved. He is perhaps most commonly remembered by the Pythagorean Theorem: a2 + b2 = c2 He declared numbers began at 3 (the triangle) and 4 (the square). Adding to those 1 and 2 – the parents of numbers – gave 10. 10 was the great number of all things, the archetype of the universe. To him, the quality of a numerical value was considered over and above the arithmetic quantities. He also regarded the five regular symmetric solids (Platonic Solids) with the greatest importance. One note: Pythagoras, Plato, Aristotle and Euclid did not have written numerals. They used small stone spheres of pebbles to do their math. We will cover the Pythagorean philosophy and Pythagorean mathematics in greater detail in subsequent articles. We will now turn back to the Quadrivium. The Quadrivium: Number, Geometry, Harmony, Cosmology Number (Arithmetic) “Arithmetic, as it was studied in the ancient world, is hardly recognizable as a relation of the subject applied today to the torture of schoolchildren…The old method of teaching arithmetic, by showing the relationship of numbers to shapes, was designed to illustrate basic principles and to lead the minds of children into the habits of reason.” John Michell A whole number is called a ‘quanta’. The study of quanta deals with: factors, ratios, triangular, square and cubic numbers, prime and perfect numbers, and the way numbers appear in sequences such as the Fibonacci and Lucas sequences. There are three levels to Number. These are: The materially numbered

The mathematician’s number (indefinite)

The Ideal or Archetypal Number (complete at 10) Architect Paul Jacques Grillo, in 1960, commented: “The world around us is a world of numbers – numbers that spell life and harmony. They are organized by the geometry of figures, all related to one another according to a sublime order, into dynamic symmetry. Glimpses into this magnificent kingdom form the basis of all our knowledge and it seems that in this domain the ancient civilizations had gone further than modern science.” John Michell writes, “In the writings of the old philosophers, there is common agreement that the true purpose of number is for investigating the universe.” “The Pythagoreans held number in the very highest regard. In fact they held that it was the nearest the human mind could get to participating with the Divine Mind. Be that as it may, the impulse that grew out of the Pythagorean community eventually gave birth to the Platonic and Socratic doctrines, those that saw mathematics, psychology and cosmology as completely integral – each being a reflection of the other: what today we might call ‘aspects of universal consciousness’.”3 As far as number and arithmetic is concerned, it is the one subject we know of in which everyone consents to its nature and behavior. It is an unambiguous universal language – one that we can all agree on. However, the symbolic level of number will differ according to times, cultures and understanding. Geometry “Sacred Geometry charts the unfolding of number is space. It differs from mundane geometry purely in the sense that the moves and concepts involved are regarded as having symbolic value, and thus, like good music, facilitates the evolution of the soul.” Miranda Lundy Geometry is the order of space as ‘Number in Space’. It unfolds in four stages: The non-dimensional point (∞)

The line (1D)

The plane (2D)

Volume (3D)

Plato, Pythagoras and those before him used pebbles to count, as there was no developed number system. As they ‘counted’ they created geometry. As Keith Critchlow says, “Arithmetic is geometry in pebbles.”

Using only perfect polygons (perfect meaning all sides and angles equal) geometry consists of: Three regular grids

triangle, square, hexagon

Five regular solids (Platonic Solids)

tetrahedron, octahedron, cube, icosahedron, dodecahedron

Eight semi-regular grids

hexagon/triangle/cube; hexagon/cube/dodecagon; triangle/cube (1 & 2); triangle/hexagon (1 & 2); triangle/dodecagon; octagon/cube

Thirteen semi-regular solids (Archimedean Solids) and their duals (13 Catalan Solids)

snub tetrahedron, truncated octahedron, cube octahedron, truncated cube, rhombicuboctahedron, great rhombicuboctahedron, snub cube, truncated icosahedron, icosidodecahedron, truncated dodecahedron, rhombicosidodecahedron, great rhombicosidodecahedron, snub dodecahedron

Incidentally, 3, 5, 8, and 13 are all Fibonacci numbers. “In the hierarchy of arts, geometry ranks equal to music, expressing through visible shapes the same numerical proportions as musical harmonies present to the ear…These proportions were not derived directly from music, but rather from a common source in number as a symbol of the divine Creation.” John Michell

Harmony (Music) – “The Nature of the Soul” “When the Soul is in harmony with itself we call it ‘peace of mind’.” – Keith Critchlow Harmony is ‘Number in Time’. It refers to frequency and harmonics. Robert Lawlor tells us the Laws of Harmonics are “considered to be universals which define the relationship and interchanges between the temporal movements and events of the heavens and the spatial order and development of the earth.” Simple ratios exist between periods or frequencies: 1:1 – unison

2:1 – the octave

3:2 – the fifth

4:3 – the fourth

The frequency of the fifth differs from that of the fourth as 9:8 – the value of the tone which gives rise to the scale. Harmony displays four musical scales: pentatonic – 5 notes (white keys of the piano A C D E G)

diatonic – 7 notes (white keys of the piano A B C D E F G)

chromatic – 12 notes (white and black keys of the piano C C# D D# E F F# G G# A A# B)

shruti – 22 notes The key to harmonic ratios is hidden in the famous Pythagorean tetractys, or pyramid of 10 dots. The tetractysis made up of the first four numbers – 1, 2, 3, 4 – which in their proportions reveal the intervals of the octave, the diapente (perfect fifth), and the diatessaron (perfect fourth). Pythagoras discovered that the seven modes, or keys, of the Greek system of music had the power to incite or allay the various emotions. As Manly P Hall wrote, “Harmony is a state recognized by great philosophers as the immediate prerequisite of beauty. A compound is termed beautiful only when its parts are in harmonious combination. The world is called beautiful and its Creator is designated the Good because good perforce must act in conformity with its own nature; and Good acting according to its own nature is harmony, because the good which it accomplishes is harmonious with the good which it is. Beauty is harmony manifesting its own intrinsic nature on the world of form.” Cosmology (Astronomy) Cosmology or Astronomy is ‘Number in Space & Time’. The word ‘cosmos’ was originated by Pythagoras which means ‘order’ and ‘adornment’. Order meaning “the arrangement of things in relation to each other according to a particular sequence, pattern or method.” Adornment meaning “adding grace, beauty, or honor.” “The Pythagoreans viewed the visible heavens as an ‘adornment’ of pure principles, the number of visible planets relating to the principles of proportional harmony. The study of the perfection of the heavens was a way of perfecting the movements of one’s own soul.” Keith Critchlow

The solar system was traditionally studied. The Pythagoreans found profound harmony in the geometry and mathematical ratios of the orbits of the sun, the planets, and their respective moons and the lessons they offered for spiritual evolution of human beings. We will cover all four of these Quadrivium studies in detail as we move through Cosmic Core. “Every diagram and system of number and every combination of harmony and the agreement of the revolution of the stars must be made manifest as one in all to him who learns in the proper way, and will be made manifest if a person learns aright by keeping his mind’s eye on unity; for it will be manifest to us as we reflect, that there is one natural bond linking all things.” Plato, Epinomis

Plato and the Search for Truth Plato, the most well-known spokesman for Pythagorean teachings (through Socrates) tells us that our world is a relative world – one that is constantly changing or ‘becoming’. “Plato introduces in his works the mediating importance of ‘soul’ or that part of our make-up that not only bears life but is the bridge between the sensible [physical] and the intelligible [spirit]. Thus the relation between the certain and uncertain is the domain of the ‘Soul’. And the Soul is a proportional harmony.”4 This is why Music – or Harmony – is called the ‘Nature of the Soul’ by Plato. “The opening discussion of Plato’s dialogue The Timaeus puts forward two vitally important concepts. First, that of extending ourselves to the utmost to find truth and secondly, the acknowledgment of the likelihoods rather than finalities in our seeking of the truth.”5 “The Pythagorean ethos of the necessity of a simultaneous pursuit of the True, the Beautiful, and the Good was profoundly and integrally transmitted in the words of Plato and becomes a reminder, when dwelt upon, that there are at least three fundamental aspects of our nature that require nourishment. Thus there is an insistence upon simultaneous concern for all three values to ensure a balanced diet. Our minds require Truth, our values require Beauty, and our wholeness, or soulness, requires the ultimate Good…Three values that are ever true and life-supporting.”6 https://en.wikipedia.org/wiki/Trivium Michell, John, The Dimensions of Paradise, Inner Traditions, 2nd edition, 2008, pg. 49 Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011 ibid. ibid. ibid.

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