Sympathetic Vibratory Physics

"The Tradition of those who preceded us on the Way of Wisdom, tells us that all things proceed from the Four Elements, and that these Four Elements are the foundation of all. They are Earth, Water, Air and Fire respectively. The Alchemists knew how to draw two Principles from these Four Elements, male and female respectively, and a third – Neuter – Principle.
These are Philosophical Sulfur, Philosophical Salt, and Philosophical Mercury. Thus, by means of a simple and salutary Operation, the Masters tell us, the Four are reduced to three. But Philosophical Sulfur, Mercury and Salt only constitute an intermediary aspect of the evolution of our Elements. From their production are born two Principles, superior to all the others. These are Wise Sulfur and Wise Mercury. Here are in reality our two supreme Arcana of the Art, and it is in their ultimate copulation that the Chrysopage will finally be born.
This Tetractys was well known to the students of Pythagoras the Wise and the Holy Saints of God, Who poured out the knowledge, nor were they ignorant of the use of His Holy Names.
Such that it comprises the whole of the key of our Alchemy." [Practical Kabbalah by R. Ambelain]

Albert Gallatin Mackey
Tetractys. The Greek word τετρακτύς 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, means four,) 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 (v. 47): “But how mayst 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 decade in the four. Thus, in assembling all numbers from one to four inclusive, the whole composition makes ten,” etc.

And 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 the tetractys, that it is said 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:
Ναὶ μὰ τὸν ἁμετέραν ψυχὴν παραδόντα τετρακτύν
Παγὰν ἁενάου φύσεως, ἀλλ' ἔρρει δι' ἔργον.
i.e.,
“I swear it by him who has transmitted into our soul the sacred tetractys,
The source of nature, whose course is eternal.”
Jamblichus gives a different phraseology of the oath, but with substantially the same meaning. 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. Dr. 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 duad, representing the number two, and answers to the potential line which, consisting of length without breadth, is bounded by two extreme points. The three following points are called the triad, representing the number three, and may be considered as having an indissoluble relation to all superficies, which consist of length and breadth, when contemplated as abstracted from thickness.” [An encyclopaedia of freemasonry and its kindred sciences, c. 1924 by Albert Gallatin Mackey]

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The Hermetic Tetractys