It was an auspicious day for the student at Crotona when Pythagoras received him into his own dwelling and welcomed him as a disciple. The candidate could now look back upon his eight years of probationary discipline with gratitude, for he knew that they had prepared him for the study of Nature’s hidden secrets and placed him on the Path leading to Adeptship.

Pythagoras began his instructions by establishing certain universal principles, proceeding from them into particulars. The key to the whole Pythagorean system, irrespective of the particular science to which it is applied, is the general formula of unity in multiplicity, the idea of the One evolving and pervading the many. This is commonly known as the Doctrine of Emanations. Pythagoras called it the Science of Numbers.

Pythagoras taught that this science—the chief of all in occultism—was revealed to men by “celestial deities,” those godlike men who were the Divine Instructors of the Third Race. It was first taught to the Greeks by Orpheus, and for centuries made known only to the “chosen few” in the Mysteries. Just before the Mysteries began to degenerate, Pythagoras instituted this teaching in his School, thus preserving under the name of “philosophy” the ancient science which, as Plato truly says, is “the greatest good that was ever imparted to men.” In his Life of Pythagoras, Iamblichus repeats the statement of Plato that the study of the science of Numbers tends to awaken that organ in the brain which the ancients described as the “eye of wisdom”—the organ now known to physiology as the pineal gland. Speaking of the mathematical disciplines, Plato says in the Republic (Book VII), “the soul through these disciplines has an organ purified and enlightened, an organ better worth saving than ten thousand corporeal eyes, since truth becomes visible through this alone.”

The present mode of teaching mathematics does little to arouse the higher mind. Even geometry, although based on the Elements of Euclid, is studied only for the purpose of acquiring a knowledge of the other parts of mathematics dependent upon it,

…without having even a dreaming perception of its first and most essential use, that of enabling its votary, like a bridge, to pass over the obscurity of a material nature, as over some dark sea, to the luminous regions of perfect reality. (Thomas Taylor: Theoretic Arithmetic of the Pythagoreans.)

In the seventh book of the Republic Plato indicates the possibilities lying behind the knowledge of numbers. He would make it compulsory for those who manage the affairs of state to study mathematics, “not in a common way, but till by intelligence itself they arrive at the survey of the nature of numbers.” This science, he assures us, should not be used merely for buying and selling, but “for facility in the energies of the soul itself.”

The Pythagorean student approached the science of mathematics from the universal point of view. By applying mathematics to both the Macrocosm and the Microcosm he was able to grasp the secrets of evolution in their minutest details. Quoting from the Neo-Pythagorean Moderatus, Porphyry says that the numerals of Pythagoras were “hieroglyphic symbols, by means whereof he explained ideas concerning the nature of things,” or the origin of the universe.

Plato, summarizing the Pythagorean formula, says that “Deity geometrizes.” The universe evolves from within outward. From the “point” a radiation equal in all directions begins, establishing a circumference, or sphere, within which all activities of the “point” are confined. The point, extending horizontally, becomes a diameter dividing the sphere into positive and negative hemispheres—the basis for action and reaction. The vertical extension of the point into a line crossing the horizontal makes the cross within the circle, and so on ad infinitum. The eleventh Chapter of The Bhagavad-Gita is a dissertation on the Pythagorean Science of Numbers, couched in Eastern terminology. There Krishna shows Arjuna the “vital geometry” of his Divine Form, with all the living lines of force therein and the countless lesser forms produced by them, representing the powers and elements that go to make up the universe.

Pythagoras described the indivisible Unity lying behind all manifestation as “No Number,” in this way repeating the statement in the Stanzas of Dzyan that “there is neither first nor last, for all is one: number issued from no number.” The plane above, therefore, can be indicated only by the nought or Circle, which Pythagoras said is the most appropriate symbol of Divinity.

On the plane below, the Monad or first number appears, and from this number the geometry of the universe emerges. Pythagoras called the Monad, or One, the first odd and therefore divine number. It is through the misinterpretation of the Pythagorean Monad that the various “personal Gods” of the different religions arose, most of whom are represented as a Trinity. In the phenomenal world the Monad becomes the apex of the manifested equilateral triangle, or the “Father.” The left line of the triangle becomes the Duad or “Mother.” This represents the origin of all the contrasts in nature, the point at which the roads of good and evil bifurcate. This being the case, the Pythagoreans are said to have “hated” the Binary. Considering the number Two as a representation of the law of polarity, they stressed its positive aspect by entering a temple on the right side and by putting on the right shoe first. The right line of the triangle represents the “Son,” described in every ancient cosmogony as one with the apex or “Father.” The line at the base of the triangle stands for the universal plane of productive nature, in which “Father-Mother-Son” are unified on the phenomenal plane as they were united in the supersensuous world by the apex.

The triangle is the most profound of all geometrical symbols. As a cosmic symbol representing the Higher Trinity of the universe it became the root of the word Deity. The ancient Greeks called the letter D (the triangular delta) “the vehicle of the Unknown Deity.” The Boeotians wrote the word Zeus with a delta, from which came the Latin Deus. The triangle is also a basic form in Nature. When the molecules of salt deposit themselves as a solid, the first shape they assume is that of a triangle. A flame is triangular in shape; hence, the word pyramid from the Greek pyr, or fire. The triangle is also the form assumed by the pine, the most primitive tree after the fern period.

The Pythagoreans called the number Four the “Key-bearer of Nature.” As a cosmic symbol it represents the universe as chaotic matter before being informed by Spirit. The cross made by the intersection of the vertical line of Spirit and the horizontal line of matter represents spiritual man crucified in the flesh, while the four-pointed star is a symbol of the animal kingdom.

The five-pointed star, the pentacle, is the symbol of man, not only of the physical man with his four limbs and head, but also of conscious, thinking man, whose fifth principle is Manas. The Pythagoreans associated the number Five with the fifth element, Ether. They called Five the “beam of the balance,” which suggests the power of choice and perhaps the final “moment of choice” for our humanity in the middle of the Fifth Round.

The number six illustrates the six directions of extension of all solid bodies. The interlaced triangles picture the union of spirit and matter, male and female. The Pythagoreans considered this number as sacred to Venus, since “the union of the two sexes, and the spagyrisation of matter by triads are necessary to develop the generative force … which is inherent in all bodies.” (Rayon:Potency of the Pythagorean Triangles.)

Pythagoras called seven a perfect number, making it the basis for “Music of the Spheres.” Regarding seven as a compound of three and four, he gave a twofold account of its meaning: On the noumenal plane the triangle is Father-Mother-Son, or Spirit, while the quaternary represents the ideal root of all material things; applied to man, the triangle represents his three higher principles, immortal and changeless, while the quaternary refers to the four lower principles which are in unstable flux. Seven not only governs the periodicity of the phenomena of life on the physical plane, but also dominates the series of chemical elements, as well as the world of sound and color, as shown by the spectroscope.

The Pythagoreans called the number eight “Justice.” In that symbol we find an expression of the eternal spiral motion of cycles, the regular inbreathing and outbreathing of the Great Breath. They called the number nine the “Ocean” and the “Horizon,” as all numbers are comprehended by and revolve within it. If we consult the Table of the Yugas on page 125 of The Ocean of Theosophy,we shall observe that all the figures may be resolved into the number nine.

Ten, or the Decade, brings all these digits back to unity, ending the Pythagorean table. In both the Microcosm and the Macrocosm the three higher numbers of the Decade stand for the invisible and metaphysical world, while the lower seven refer to the realm of physical phenomena.

The Tetraktys of Pythagoras—composed of ten dots arranged in four rows to form a triangle—was the sacred symbol upon which the Pythagoreans took their most binding oath:

“I swear by him who the Tetraktys found,
Whence all our wisdom springs and which contains
Perennial Nature’s fountain, cause and root.”

Theon of Smyrna says that this symbol was honored by the Pythagoreans “because it appears to contain the nature of all things.” H.P.B. indicates the extraordinary philosophical value of theTetraktys in The Secret Doctrine (I, 612). According to Iamblichus, the Pythagorean Tetraktys had eleven forms, each one applying to some one particular phase of cosmic or terrestrial life.

Pythagoras applied the Science of Numbers to music, giving the Western world the mathematical basis of its present musical system. The abstract Circle of music is Sound. The mathematical point within that circle, from which the music of our earth emerges, is the “Tone of Nature,” called Kung by the ancient Chinese. The “line” of music, derived from the ratio 2:3, is what is now called the “perfect fifth.” The rotation of this line forms the “Circle of Fifths,” which gives the basis of all key relationships.

The music of this planet, according to Pythagoras, is but a small copy of the “Music of the Spheres.” The seven tones of the musical scale correspond to the seven sacred planets, each of which is characterized by a certain tone. As Shakespeare makes Lorenzo say in The Merchant of Venice, “There’s not the slightest orb which thou beholdest but in its motion like an angel sings.” The study of music was obligatory in the Pythagorean School, not only as a science but also as a healing agent. Iamblichus informs us that “Pythagoras believed that music greatly contributed to health, if it was used in the proper manner.” Pythagoras taught that the purest type of sound comes from stringed instruments and that wind instruments tend to excite the lower nature rather than to quiet it, an observation later corroborated by Plato.

The study of astronomy was a duty of the School. Pythagoras taught the heliocentric system and the sphericity of the earth; he declared that the moon is a dead planet which receives its light from the sun and described the composition of the Milky Way. More than a thousand years later both Bruno and Galileo derived their theories of astronomy from Pythagorean fragments.

The esoteric students of Pythagoras were given the Mystery teachings in regard to the nature of the soul, its relation to the body and its ultimate destiny. Pythagoras taught that the soul of man is derived from the World-Soul; hence is immortal and cannot be destroyed by death. The soul of man, he said, accomplishes its evolution by means of numberless incarnations on earth. He frequently spoke to his pupils about their own former lives, and when asked about himself said that he had come into the region of mortality to benefit mankind. He also taught the doctrine of Karma, saying that all the seeming injustices on earth are explained by the fact that every life on earth is but a reward or punishment for deeds performed in previous lives. No outside circumstances are to blame for our unhappy lives, he said, since “men draw upon themselves their own misfortunes, voluntarily and of their own free choice.”

Applying the Science of Numbers to the problem of good government, Pythagoras first made himself a “point” in which great spiritual forces were focused, and from that “point” the radii of their influence extended. The Pythagorean School eventually became a small model city, its form of government being adopted by Crotona. From Crotona the sphere of Pythagorean influence expanded to include the neighboring towns, where legislative systems based upon Pythagorean principles lasted for generations.

When Pythagoras was almost a hundred years old he went to Delos to attend the funeral ceremonies of an old friend. One evening, when the Teacher and forty of his pupils were talking together, some of his former pupils who had been expelled from his School set fire to the building where they were assembled, and Pythagoras, with thirty-eight of his pupils, were consumed in the flames.

After the death of the Teacher the School at Crotona was closed and the students departed from Italy. Fearing that the very word philosophy — a word which Pythagoras had coined—would disappear from the Greek language, some of these loyal disciples collected the writings of the older Pythagoreans and wrote down many things which Pythagoras himself had said. These writings were passed down from teacher to pupil, or from father to son, for many generations.

The direct successor to Pythagoras—if such a man could be said to have a successor—was his pupil Aristæus. After him came Pythagoras’ son Mnesarchus, who was named after his grandfather. The Pythagorean fragments were preserved by two hundred and thirty-five of his loyal disciples, two hundred and eighteen of whom were men, the other seventeen women. At the present day all that remains of his ethical precepts is found in the Golden Verses.