“The Tetractys as the fountain of perennial nature."

— Proclus - On the Timaeus 3
The Tetractys (Greek τετρακτ?ς), or Tetrad, or the Tetractys of the Decad is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. 
The Greek word signifies, literally, the number four, and is therefore synonymous with the quaternion; but it has been peculiarly applied to a symbol of the Pythagoreans, which is composed of ten dots arranged in a triangular form of four rows. 
This figure was in itself, as a whole, emblematic of the Tetragrammaton, or sacred name of four letters, for Tetractys, in Greek, and was undoubtedly learned by Pythagoras during his visit to Babylon. But the parts of which it is composed were also pregnant symbols. Thus the one point was a symbol of the Active Principle or Creator, the two points of the Passive Principle or Matter, the three of the world proceeding from their union, and the four of the liberal arts and sciences, which may be said to complete and perfect that world. 
This arrangement of the ten points in a triangular form was called the Tetractys or number four, because each of the sides of the triangle consisted of four points, and the whole number of ten was made up by the summation of the first four figures, 1 + 2 + 3 +4 = 10. 
Hierocles says, in his Commentaries on the Golden Verses: 
"But how comes God to be the Tetractys? This thou mayst learn in the sacred book ascribed to Pythagoras, in which God is celebrated as the number of numbers. For if all things exist by His eternal decrees, it is evident that in each species of things the number depends on the cause that produces them. Now the power of ten is four; for before we come to a complete and perfect decade, we discover all the virtue and perfection of the ten in the four. Thus, in assembling all numbers from one to four inclusive, the whole composition makes Ten."
Dacier, in his notes on these Commentaries and on this particular passage, remarks that: 
"Pythagoras, having learned in Egypt the name of the true God, the Mysterious and Ineffable name Jehovah, and finding that in the original tongue it was composed of four letters, translated it into his own language by the word Tetractys, and gave the true explanation of it, saying that it properly signified the source of nature that perpetually rolls along." 
So much did the disciples of Pythagoras venerate Tetractys, that it is said that they took their most solemn oaths, especially that of initiation, upon it. The exact words of the oath are given in the Golden Verses, and are referred to by Jamblichus in his Life of Pythagoras. 
"I swear it by Him who has transmitted into our soul the sacred Tetractys the source of nature, whose course is eternal." 
Iamblichus gives a different phraseology of the oath, but with substantially the same meaning. In the symbols of Freemasonry, we will find the sacred Delta bearing the nearest analogy to the Tetractys of the Pythagoreans. 
The outline of these points form, it will be perceived, a triangle; and if we draw short lines from point to point, we will have within this great triangle nine smaller ones. 
Doctor Hemming, in his revision of the English lectures, adopted in 1813, thus explains this symbol: 
"The great triangle is generally denominated Pythagorean, because it served as a principal illustration of that philosopher's system. This emblem powerfully elucidates the mystical relation between the numerical and geometrical symbols."
It is Composed of ten points so arranged as to form one great equilateral triangle and at the same time to divide it into nine similar triangles of smaller dimensions. The first of these, representing unity, is Called a Monad, and answers to what is denominated a point in geometry, each being the principle by the multiplication of which all Combinations of form and number are respectively generated. 
The next two points are denominated a Decad, representing the number two, and answers to the geometrical line which, consisting of length without breadth, is hounded by two extreme points. The three following points are called the triad, representing the number three, and may be considered asks having an indissoluble relation to all superficies which consist of length and breadth, when Contemplated as abstracted from thickness. 
Doctor Hemming does not appear to have improved on the Pythagorean symbolization. 
"Ten, or a group of ten; a sacred number because the universe is built on the model of the Decad, the individual and the universe as a whole being tenfold though Septenary in manifestation. The One or cosmic Monad is sometimes spoken of as emanating the other nine, and by including the One itself we get the Ten rays of the Logos, the Sephiroth, etc., which are spoken of as seven in the manifested universe."
The Decad may be considered as a double five or a three triangles and a unity. It is represented in ancient Greece by the Pythagorean Tetraktys, of which the three upper dots represent the unmanifest universe, and the lower seven the manifest. The Decad is the radix of the denary scale of notation derived from ancient India.
As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music.
The First Logos is the real Pythagorean Tetractys. It is the Esoteric Logos of the Invisible Monad.
The Tetrad is esteemed in the Kabbalah, as it was by Pythagoras, the most perfect, or rather sacred number, because it emanated from the one, the first manifested Unit, or rather the three in one. Yet the latter has been ever impersonal, sexless, incomprehensible, though within the possibility of the higher mental perceptions. 
Tetragrammaton is Logos Demiourgos; it is the Tetractys of the profane.
The first manifestation of the eternal Monad was never meant to stand as the symbol of another symbol, the UNBORN for the Element-born, or the one LOGOS for the Heavenly man. Tetragrammaton, or the Tetractys, of the Greeks, is the Second Logos, the Demiourgos. 
The Tetrad, as Thomas Taylor thought, is the animal itself of Plato, who, as Syrianus justly observes, was the best of the Pythagoreans; it subsists at the extremity of the intelligible triad, as is most satisfactorily shown by Proclus in the third book of his treatise on the theology of Plato. And between these two triads (the double triangle), the one intelligible, and the other intellectual, another order of gods exists which partakes of both extremes.
The Pythagorean world consisted of a double quaternary: one, of the pure, intellectual world (Mahat), and another of the sensible world of matter (Pythagoras Kosmos, our beautiful world).
The Tetraktys is the number four or a group of four, a Tetrad or quaternary. 
The Tetraktys of Pythagoras, as an emblem, consisted of a triangle formed by ten dots, of which he says: 
“In what you conceive as four there are ten; then, a perfect triangle and the Tetraktys (four) make seven.”
The highest group of Dhyani-Chohans within the divine realm and Government of Fire, though divided into three classes, is synthesized by the One, thus making Four or Tetractys.
Madame Blavatsky in her Commentary on the Pistis Sophia says: 
"The First Tetractys — 1) Arrhetos (ineffable) with 7 elements; 2) Sige (silence) with elements; Pater (father) with five elements; and 4) Aletheia (truth) with 7 elements, for a total of 24 elements." 
"The Second Tetractys — 1) Logos (word) with 7 elements; 2) Zoe (life) with five elements; Anthropos (man) with five elements; and 4) Ekklesia (assembly) with 7 elements, for a total of 24 elements, which together with Christos gives a total of 49 elements."
The Pythagoreans believed that the Tetractys was an illustration of existence, of the way everything is structured in the universe. Meditating upon the Tetractys symbol is believed to signify the desire to seek Divine understanding and a yearning to attain a close relationship with God.
From the contemplation of the order and progression of the Numbers out of the Monad may be discovered the true relationship between natures and their Cause. 
He who understands this mystery is of all men the wisest. 
Numbers are the keys to the flow of Universal Energy.
The Tetractys or Tetrad, or the Tetractys of the Decad is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music.
The first four numbers symbolize the musica universalis and the Cosmos as:
1) Monad – Unity.
2) Dyad – Power – Limit/Unlimited (Peras / Apeiron).
3) Triad – Harmony.
4) Tetrad – Kosmos.
The four rows add up to ten, which was unity of a higher order (The Dekad).
The Tetractys symbolizes the four classical elements — air, fire, water, and earth.
The Tetractys represented the organization of space.
The first row represented zero dimensions (a point).
The second row represented one dimension (a line of two points).
The third row represented two dimensions (a plane defined by a triangle of three points).
The fourth row represented three dimensions (a Tetrahedron defined by four points).
A prayer of the Pythagoreans shows the importance of the Tetractys (sometimes called the "Mystic Tetrad"), as the prayer was addressed to it.
"Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all."
As a portion of the secret religion, initiates were required to swear a secret oath by the Tetractys. They then served as novices, which required them to observe silence for a period of five years.
The Pythagorean oath also mentioned the Tetractys:
"By that pure, holy, four lettered name on high, nature's eternal fountain and supply, the parent of all souls that living be, by him, with faith find oath, I swear to thee."
It is said that the Pythagorean musical system was based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales.
That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1. Note that the Diapason, 2:1 (octave), and the Diapason plus Diapente, 3:1 (compound fifth or perfect twelfth), are consonant intervals according to the Tetractys of the Decad, but that the Diapason plus Diatessaron, 8:3 (compound fourth or perfect eleventh), is not.
The Tetractys (also known as the Decad) is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself — in seed-like form — the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the mysteries of the divine realm. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity.
In the work by anthropologist Raphael Patai entitled The Hebrew Goddess, the author argues that the tetractys and its mysteries influenced the early Kabbalah. A Hebrew Tetractys has the letters of the Tetragrammaton inscribed on the ten positions of the Tetractys, from right to left. It has been argued that the Kabbalistic Tree of Life, with its ten spheres of emanation, is in some way connected to the Tetractys, but its form is not that of a triangle. 
The occultist Dion Fortune writes:
"The point is assigned to Kether. The line to Chokmah. The two-dimensional plane to Binah. Consequently the three-dimensional solid naturally falls to Chesed."
The relationship between geometrical shapes and the first four Sephirot is analogous to the geometrical correlations in Tetraktys, shown above under Pythagorean symbol, and unveils the relevance of the Tree of Life with the Tetraktys.
In a Tarot reading, the various positions of the tetractys provide a representation for forecasting future events by signifying according to various occult disciplines, such as Alchemy. 
If Pythagoras found himself transported to the modern world, he would have much to learn about technology, science, and human thought. But is there something Pythagoras can still teach us today in his symbol of the Tetractys? What do the dots reveal? How is it significant to Freemasonry?
To start with, Tetractys refers to a symbol of the Pythagoreans which consists of four rows of dots containing one, two, three, and four dots respectively which form an equilateral triangle. Many have found the tetractys full of sublime meaning.
Pythagoras was born in Samos in the sixth century B.C., Pythagoras was both a mystic and a scientist, although some scholars tend to praise his mathematical prowess while looking away with embarrassment at his perceived “mysticism.” For Pythagoreans, they were one and the same.
The Science of Number was the cornerstone of the Pythagoreans. It describes, if not yet everything, at least something very important about physical reality, namely the sizes and shapes of the objects that inhabit it.
For them the whole of the Universe, metaphysical and material, was contained within, and could be expressed and described by the digits of Number 10, the Pythagorean Decad.
This Decade representing the Universe and its evolution out of Silence and the unknown Depths of the Spiritual Soul, or Anima Mundi, presented two sides or aspects to the student. It could be, and was at first so used and applied to the Macrocosm, after which it descended to the Microcosm, or Man. 
There was, then, the purely intellectual and metaphysical, or the “inner Science,” and the as purely materialistic or “surface science,” both of which could be expounded by and contained in the Decad. 
The Pythagoreans influenced the world by the simple expression:
"All is number." 
? Pythagoras
What Did Pythagoras mean by this famous motto “All is Number?”
Is it possible to listen to this message today afresh, with Pythagorean ears? 
Manly Palmer Hall dedicated an entire chapter in his work “The Secret Teachings of all Ages” to the both mystical and philosophical qualities of Pythagorean numbers.
Manly Palmer Hall wrote:
"The ten dots, or Tetractys of Pythagoras, was a symbol of the greatest importance, for to the discerning mind it revealed the mystery of universal nature."
Mr. Hall states that if one examines the tetractys symbolically a wealth of otherwise hidden wisdom begins to reveal itself.
The Prestonian Lectures (1772) give us further insight into some of the possible masonic thinking on the tetractys in the 1800’s. It was the subject in one of the series of lectures written by Brother William Preston for instruction and education of the Lodge members.
An excerpt of the Lecture (1772) goes as follows:
The Pythagorean philosophers and their ancestors considered a Tetractys or No. 4:
1st as containing the Decad.
2nd as completing an entire and perfect triangle.
3rd as comprising the 4 great principles of arithmetic and geometry.
4th as representing in its several points the 4 elements of Air, Fire, Water and Earth, and collectively the whole system of the universe.
Lastly as separately typifying the 4 external principles of existence, generation, emanation, creation and preservation, thence collectively denoting the Great Architect of the Universe Wherefore to swear by the Tetractys was their most sacred and inviolate oath.
In other words, it is taught to Freemasons that a four-fold pattern permeates the natural world, examples of which are the point, line, surface and solid and the four elements earth, water, air and fire. Musically they represent the perfect consonants: the unison, the octave, the fifth and the fourth.
The Divine Creator in Freemasonry is sometimes referred to as The Great “Architect” or Grand “Geometrician” always building the universe through the creative tools of the geometer. Tetractys itself can be interpreted as a divine blueprint of creation. 
Some say that Pythagoras and his successors had two ways of teaching, one for the profane, and one for the initiated. The first was clear and unveiled, the second was symbolic and enigmatic. In order to achieve mastery of this universe, a person has to discover the veiled meaning of numbers hidden in all things.
I have often wondered if we could hypothetically peer into the mind of the Grand Geometrician, and the veil was lifted, what design would we see?
The Grand Design:
Perhaps we would see how the Master Builder has ordered all things by measure and number and weight. Throughout the structure of the universe the properties of number are manifested. Geometry is fundamental to the work of the Masonic builders. It is engaged with the first configurations of the Plan upon which the form is erected and the idea materialized.
Examining numbers symbolically, they represent more than quantities; they also have qualities. Madame Blavatsky in the “Secret Doctrine” tells us the numbers are entities. They are mysterious. They are essential to all forms. They are to be found in the realm of essential consciousness. They are clues to our evolution.
Madame Blavatsky emphasizes that the study of numbers is not only a way of understanding nature, but it is also a means of turning the mind away from the physical world which Pythagoras held to be transitory and unreal, leading to the contemplation of the “real.”
Personally, I find that the Masonic teachings in all their many symbolic forms is a good way to study numbers. The reason I continually come back to Pythagorean philosophy is the tradition of music theory. In music, the Divine patterns of the Grand Geometrician are expressed in musical ratios. Harmony through sound, therefore, can be applied to all phenomena of nature, even going so far as to demonstrate the harmonic relationship of the planets, constellations, elements and everything, really. The reason being that all life vibrates, like the string. 
Why do Freemasons connect the dots? Like many symbols, the Tetractys can lead a craftsman down a rabbit hole of self-discovery. By rabbit hole, I mean a portal into a mysterious and infinite wonderland of formulas filled with beauty, confusion and intrigue – a place to encounter all sorts of adventures with concepts beyond our wildest dreams that keeps us coming back for more.
This represents four stages of evolution: a Monad, a dual creative force or Duad, the world of forms, and the world of complete and concrete manifestation. This arrangement of dots enables one to deduce any of the numbers from 1 to 10. It was held in such high esteem by the Pythagoreans that their most binding oath was made upon the Tetraktys.
“It has a very mystic an varied signification. First of all it is Unity, or the ‘One’ under four different aspects; then it is the fundamental number Four, the Tetrad containing the Decad, or Ten, the number of perfection, finally it signifies the primeval Triad (or Triangle) merged in the divine Monad. The mystical Decad, the resultant of the Tetraktys, or the 1 + 2 + 3 + 4 = 10, is a way of expressing this idea. One is the impersonal principle ‘God’; the Two, matter; the Three, combining Monad and Duad partaking of the nature of both, is the phenomenal world; the Tetrad, or form of perfection expresses the emptiness of all; and the Decad, or sum of all, involves the entire Kosmos." 
? Helena Petrovna Blavatsky - Collected Writings
"The more deeply we study the processes of nature the greater in every direction becomes our admiration for the wonderful work of Him who made it all."
? Charles Webster Leadbeater
#Tetraktys

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