(11) To Pythagoras music was one of the dependencies of the divine science of mathematics, and its harmonies were inflexibly controlled by mathematical proportions. The Pythagoreans averred that mathematics demonstrated the exact method by which the good established and maintained its universe. Number therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. After discovering these harmonic ratios, Pythagoras gradually initiated his disciples into this, the supreme arcanum of his Mysteries. He divided the multitudinous parts of creation into a vast number of planes or spheres, to each of which he assigned a tone, a harmonic interval, a number, a name, a color, and a form. He then proceeded to prove the accuracy of his deductions by demonstrating them upon the different planes of intelligence and substance ranging from the most abstract logical premise to the most concrete geometrical solid. From the common agreement of these diversified methods of proof he established the indisputable existence of certain natural laws.

(53) Timaeus: Now all triangles derive their origin from two triangles, each having one angle right and the others acute ; and the one of these triangles has on each side half a right angle marked off by equal sides, while the other has the right angle divided into unequal parts by unequal sides. These we lay down as the principles of fire and all the other bodies, proceeding according to a method in which the probable is combined with the necessary; but the principles which are still higher than these are known only to God and the man who is dear to God.

(24) There is no doubt that the Tabernacle, its furnishings and ceremonials, when considered esoterically, are analogous to the structure, organs, and functions of the human body. At the entrance to the outer court of the Tabernacle stood the Altar of Burnt Offerings, five cubits long and five cubits wide but only three cubits high. Its upper surface was a brazen grill upon which the sacrifice was placed, while beneath was a space for the fire. This altar signified

(1) After this we must narrate how, when he had admitted certain persons to be his disciples, he distributed them into different classes according to their respective merits. For it was not fit that all of them should equally participate of the same things, as they were naturally dissimilar; nor was it indeed right that some should participate of all the most honorable auditions, but others of none, or should not at all partake of them. For this would be uncommunicative and unjust. While therefore he imparted a convenient portion of his discourses to each, he benefited as much as possible all of them, and preserved the proportion of justice, by making each a partaker of the auditions according to his desert.

Hence, in conformity to this method, he called some of them Pythagoreans, but others Pythagorists; just as we denominate some men Attics, but others Atticists. Having therefore thus aptly divided their names, some of them he considered to be genuine, but he ordained that others should show themselves to be the emulators of these. He ordered therefore that with the Pythagoreans possessions should be shared in common, and that they should always live together; but that each of the others should possess his own property apart from the rest, and that assembling together in the same place, they should mutually be at leisure for the same pursuits. And thus each of these modes was derived from Pythagoras, and transmitted to his successors.

Again, there were also with the Pythagoreans two forms of philosophy; for there were likewise two genera of those that pursued it, the Acusmatici, and the Mathematici. Of these however the Mathematici are acknowledged to be Pythagoreans by the rest; but the Mathematici do not admit that the Acusmatici are so, or that they derived their instruction from Pythagoras, but from Hippasus. And with respect to Hippasus, some say that he was a Crotonian, but others a Metapontine. But the philosophy of the Acusmatici consists in auditions unaccompanied with demonstrations and a reasoning process; because it merely orders a thing to be done in a certain way, and that they should endeavour to preserve such other things as were said by him, as so many divine dogmas.

They however profess that they will not speak of them, and that they are not to be spoken of; but they conceive those of their sect to be the best furnished with wisdom, who retained what they had heard more than others. But all these auditions are divided into three species. For some of them indeed signify what a thing is; others what it especially is; but others, what ought, or what ought not, to be done. The auditions therefore which signify what a thing is, are such as, What are the islands of the blessed? The sun and moon. What is the oracle at Delphi? The tetractys. What is harmony? That in which the Syrens subsist . But the auditions which signify what a thing especially is, are such as, What is the most just thing?

To sacrifice. What is the wisest thing? Number. But the next to this in wisdom, is that which gives names to things. What is the wisest of the things that are with us, [i. e. which pertain to human concerns]? Medicine. What is the most beautiful? Harmony. What is the most powerful? Mental decision. What is the most excellent? Felicity. What is that which is most truly asserted? That men are depraved. Hence they say that Pythagoras praised the Salaminian poet Hippodomas, because he sings:

(526) And indeed, you will not easily find a more difficult study, and not many as difficult. You will not. And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up. I agree. Let this then be made one of our subjects of education. And next, shall we enquire whether the kindred science also concerns us? You mean geometry? Exactly so. Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general is or is not a geometrician. Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates rather to the greater and more advanced part of geometry— whether that tends in any degree to make more easy the vision of the idea of good; and thither, as I was saying, all things tend which compel the soul to turn her gaze towards that place, where is the full perfection of being, which she ought, by all means, to behold. True, he said. Then if geometry compels us to view being, it concerns us; if becoming only, it does not concern us?

(9) Life is the great mystery, and only those who pass successfully through its tests and trials, interpreting them aright and extracting the essence of experience therefrom, achieve true understanding. Thus, the temples were built in the form of the world and their rituals were based upon life and its multitudinous problems. Nor only was the Tabernacle itself patterned according to Egyptian mysticism; its utensils were also of ancient and accepted form. The Ark

(527) Yes, that is what we assert. Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is in flat contradiction to the ordinary language of geometricians. How so? They have in view practice only, and are always speaking, in a narrow and ridiculous manner, of squaring and extending and applying and the like—they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science. Certainly, he said. Then must not a further admission be made? What admission? That the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient. That, he replied, may be readily allowed, and is true. Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Nothing will be more likely to have such an effect. Then nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. Moreover the science has indirect effects, which are not small. Of what kind? he said. There are the military advantages of which you spoke, I said; and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not. Yes indeed, he said, there is an infinite difference between them. Then shall we propose this as a second branch of knowledge which our youth will study? Let us do so, he replied.

(3) By Gematria is meant not only the exchange of letters for their numerical equivalents but also the method of determining by an analysis of its measurements the mystic purpose for which a building or other object was constructed. S. L. MacGregor-Mathers, in The Kabbalah Unveiled, gives this example of the application of Gematria: "Thus also the passage, Gen. xviii. 2 VHNH SHLSHH, Vehenna Shalisha, 'And lo, three men,' equals in numerical value 'ALV MIKAL GBRIAL VRPAL, Elo Mikhael Gabriel Ve-Raphael,' These are Mikhael, Gabriel and Raphael; 'for each phrase = 701." Assuming the sides of a scalene to be 11, 9, and 6 inches, a triangle of such dimensions would then be an appropriate symbol of Jehovah, for the sum of its three sides would be 26, the numerical value of the Hebrew word IHVH. Gematria also includes the system of discovering the arcane meaning of a word by analyzing the size and arrangement of the strokes employed in the formation of its various letters. Gematria was employed by the Greeks as well as the Jews. The books of the New Testament--particularly those attributed to St. John--contain many examples of its use. Nicephorus Callistus declared the Gospel according to St. John to have been discovered in a cavern under the Temple at Jerusalem, the volume having been secreted "long anterior to the Christian æra." The existence of interpolated material in the fourth Gospel substantiates the belief that the work was originally written without any specific reference to the man Jesus, the statements therein accredited to Him being originally mystical discourses delivered by the personification of the Universal Mind. The remaining Johannine writings--the Epistles and the Apocalypse--are enshrouded by a similar veil of mystery.

(22) [paragraph continues] My House be established. "In the splendor of mental and spiritual illumination, the High Priest stood between the pillars as a mute witness to the perfect virtue of equilibrium--that hypothetical point equidistant from all extremes. He thus personified the divine nature of man in the midst of his compound constitution--the mysterious Pythagorean Monad in the presence of the Duad. On one side towered the stupendous column of the intellect; on the other, the brazen pillar of the flesh. Midway between these two stands the glorified wise man, but he cannot reach this high estate without first suffering upon the cross made by joining these pillars together. The early Jews occasionally represented the two pillars, Jachin and Boaz, as the legs of Jehovah, thereby signifying to the modern philosopher that Wisdom and Love, in their most exalted sense, support the whole order of creation--both mundane and supermundane.

(2) At the likeness of these three, we profit on three manners in this grace of contemplation. Sometime we profit only by grace, and then we be likened unto Moses, that for all the climbing and the travail that he had into the mount might not come to see it but seldom: and yet was that sight only by the shewing of our Lord when Him liked to shew it, and not for any desert of his travail. Sometime we profit in this grace by our own ghostly cunning, helped with grace, and then be we likened to Bezaleel, the which might not see the Ark ere the time that he had made it by his own travail, helped with the ensample that was shewed unto Moses in the mount. And sometime we profit in this grace by other men’s teaching, and then be we likened to Aaron, the which had it in keeping and in custom to see and feel the Ark when him pleased, that Bezaleel had wrought and made ready before to his hands.

(16) "The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in ἀναλογία. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man. For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils [and from that point] to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown."

(4) Earlier in the same work, Plutarch also notes: "For as the power of the triangle is expressive of the nature of Pluto, Bacchus, and Mars; and the properties of the square of Rhea, Venus, Ceres, Vesta, and Juno; of the Dodecahedron of Jupiter; so, as we are informed by Eudoxus, is the figure of fifty-six angles expressive of the nature of Typhon." Plutarch did not pretend to explain the inner significance of the symbols, but believed that the relationship which Pythagoras established between the geometrical solids and the gods was the result of images the great sage had seen in the Egyptian temples.

(3) "The Pythagoreans indeed go farther than this, and honour even numbers and geometrical diagrams with the names and titles of the gods. Thus they call the equilateral triangle head-born Minerva and Tritogenia, because it may be equally divided by three perpendiculars drawn from each of the angles. So the unit they term Apollo, as to the number two they have affixed the name of strife and audaciousness, and to that of three, justice. For, as doing an injury is an extreme on the one side, and suffering one is an extreme on the on the one side, and suffering in the middle between them. In like manner the number thirty-six, their Tetractys, or sacred Quaternion, being composed of the first four odd numbers added to the first four even ones, as is commonly reported, is looked upon by them as the most solemn oath they can take, and called Kosmos." (Isis and Osiris.)

(16) The court of the Tabernacle was an enclosed area, fifty cubits wide and one hundred cubits long, circumscribed by a wall of linen curtains hung from brazen pillars five cubits apart. (The cubit is an ancient standard of measurement, its length being equal to the distance between the elbow and the extreme end of the index finger, approximately eighteen inches.) There were twenty of these pillars on each of the longer sides and ten on the shorter. Each pillar had a base of brass and a capital of silver. The Tabernacle was always laid out with the long sides facing north and south and the short sides facing east and west, with the entrance to the east, thus showing the influence of primitive sun worship.

(38) The square base of the Pyramid is a constant reminder that the House of Wisdom is firmly founded upon Nature and her immutable laws. "The Gnostics," writes Albert Pike, "claimed that the whole edifice of their science rested on a square whose angles were: Σιγη, Silence; Βυθος, Profundity; Νους, Intelligence; and Αληθεια Truth." (See Morals and Dogma.) The sides of the Great Pyramid face the four cardinal angles, the latter signifying according to Eliphas Levi the extremities of heat and cold (south and north) and the extremities of light and darkness (east and west). The base of the Pyramid further represents the four material elements or substances from the combinations of which the quaternary body of man is formed. From each side of the square there rises a triangle, typifying the threefold divine being enthroned within every quaternary material nature. If each base line be considered a square from which ascends a threefold spiritual power, then the sum of the lines of the four faces (12) and the four hypothetical squares (16) constituting the base is 28, the sacred number of the lower world. If this be added to the three septenaries composing the sun (21), it equals 49, the square of 7 and the number of the universe.

(54) Timaeus: the equilateral triangle is constructed as a third. The reason why is a longer story; but should anyone refute us and discover that it is not so, we begrudge him not the prize. Accordingly, let these two triangles be selected as those wherefrom are contrived the bodies of fire and of the other elements,— one being the isosceles, and the other that which always has the square on its greater side three times the square on the lesser side. Moreover, a point about which our previous statement was obscure must now be defined more clearly. It appeared as if the four Kinds,

(26) TABLE IX, Figure 9. Figure 9 is a synthesis of the Old and New Testaments and represents the interblending planes of being. In the right margin the seven outer circles contain the names of the planetary angels. The words in the graduated circles from the top triangle downward read: (1) Abyss of Compassion; (2) Zion; (3) The New Heaven and the New Earth; (4) The New Jerusalem; (5) Paradise; (6) The Bosom of Abraham; (7) The Outer Courts of the Lord. From below the circles of darkness reach upward, each divine principle being opposed by an infernal opposite. The small circle on the left containing a triangle and cross is named The Tree of Life, and that on the right The Tree of the Knowledge of Good and Evil. In the center of the diagram is the Trinity, joined with the superior and inferior planes by lines of activity.

(25) TABLE VII., Figure 6. In Figure 6 the letter a marks the center of eternity. The motion of the rays toward b, d, and c was the first divine manifestation and is symbolized by the equilateral triangle, b, d, c. The eternal world within the inner circle became manifest in the water (salt), the light (mercury), and the fire (sulphur) of the archetypal world, represented by the three circles (f, e, g) within the triangle of complete equality (h, i, k), which is in turn surrounded by the circle of the high throne. The circle f is named understanding; e, wisdom; g, reason. In circle i is the word Father; in circle h, Son; in circle k, Spirit. The seven outer circles are the seven spirits before the throne. The lower part of the figure is similar to Figures 53 and 54. The outer circles are the angelic world ending in the cognizable world of the Sons of God. Then comes the circle of the visible constellations and fixed stars; within this is the solar system with the sun as the center (l). Ungrund means the Abyss.

(37) In the twenty-first and twenty-second chapters are pictured the new heaven and the new earth to be established at the close of Ahriman's reign. St. John, carried in the spirit to a great and high mountain (the brain), beheld the New Jerusalem descending as a bride adorned for her husband. The Holy City represents the regenerated and perfected world, the trued ashlar of the Mason, for the city was a perfect cube, it being written, "the length and the breadth and the height of it are equal." The foundation of the Holy City consisted of a hundred and forty-four stones in twelve rows, from which it is evident that the New Jerusalem represents the microcosm, patterned after the greater universe in which it: stands. The twelve gates of this symbolic dodecahedron are the signs of the zodiac through which the celestial impulses descend into the inferior world; the jewels are the precious stones of the zodiacal signs; and the transparent golden streets are the streams of spiritual light along which the initiate passes on his path towards the sun. There is no material temple in that city, for God and the Lamb are the temple; and there is neither sun nor moon, for God and the Lamb are the light. The glorified and spiritualized initiate is here depicted as a city. This city will ultimately be united with the spirit of God and absorbed into the Divine Effulgency.

(1) Of his wisdom, however, the commentaries written by the Pythagoreans afford, in short, the greatest indication; for they adhere to truth in every thing, and are more concise than all other compositions, so that they savour of the ancient elegance of style, and the conclusions are exquisitely deduced with divine science. They are also replete with the most condensed conceptions, and are in other respects various and diversified both in the form and the matter. At one and the same time likewise, they are transcendently excellent, and without any deficiency in the diction, and are in an eminent degree full of clear and indubitable arguments, accompanied with scientific demonstration, and as it is said, the most perfect syllogism; as he will find to be the case, who, proceeding in such paths as are fit, does not negligently peruse them.

This science, therefore, concerning intelligible natures and the Gods, Pythagoras delivers in his writings from a supernal origin. Afterwards, he teaches the whole of physics, and unfolds completely ethical philosophy and logic. He likewise delivers all-various disciplines, and the most excellent sciences. And in short there is nothing pertaining to human knowledge which is not accurately discussed in these writings. If therefore it is acknowledged, that of the [Pythagoric] writings which are now in circulation, some were written by Pythagoras himself, but others consist of what he was heard to say, and on this account are anonymous, but are referred to Pythagoras as their author;—if this be the case, it is evident that he was abundantly skilled in all wisdom.

But it is said that he very much applied himself to geometry among the Egyptians. For with the Egyptians there are many geometrical problems; since it is necessary that from remote periods, and from the time of the Gods themselves, on account of the increments and decrements of the Nile, those that were skilful should have measured all the Egyptian land which they cultivated. Hence also geometry derived its name. Neither did they negligently investigate the theory of the celestial orbs, in which likewise Pythagoras was skilled. Moreover, all the theorems about lines appear to have been derived from thence. For it is said that what pertains to computation and numbers, was discovered in Phœnicia. For some persons refer the theorems about the celestial bodies to the Egyptians and Chaldeans in common.

It is said therefore, that Pythagoras having received and increased all these [theories,] imparted the sciences, and at the same time demonstrated them to his auditors with perspicuity and elegance. And he was the first indeed that denominated philosophy, and said that it was the desire, and as it were love of wisdom. But he defined wisdom to be the science of the truth which is in beings. And he said that beings are immaterial and eternal natures, and alone possess an efficacious power, such as incorporeal essences. But that the rest of things are only homonymously beings, and are so denominated through the participation of real beings, and such are corporeal and material forms, which are generated and corrupted, and never truly are.

And that wisdom is the science of things which are properly beings, but not of such as are homonymously so. For corporeal natures are neither the objects of science nor admit of a stable knowledge, since they are infinite and incomprehensible by science, and are as it were, non-beings, when compared with universals, and are incapable of being properly circumscribed by definition. It is impossible however to conceive that there should be science of things which are not naturally the objects of science. Hence it is not probable that there will be a desire of science which has no subsistence, but rather that desire will be extended to things which are properly beings, which exist with invariable permanency, and are always consubsistent with a true appellation.

For it happens that the perception of things which are homonymously beings, and which are never truly what they seem to be, follows the apprehension of real beings; just as the knowledge of particulars follows the science of universals. For he who knows universals properly, says Archytas, will also have a clear perception of the nature of particulars. Hence things which have an existence are not alone, nor only-begotten, nor simple, but they are seen to be various and multiform. For some of them are intelligible and incorporeal natures, and which are denominated beings; but others are corporeal and fall under the perception of sense, and by participation communicate with that which has a real existence. Concerning all these therefore, he delivered the most appropriate sciences, and left nothing [pertaining to them] uninvestigated.

He likewise unfolded to men those sciences which are common [ to all disciplines ,] as for instance the demonstrative, the definitive, and that which consists in dividing, as may be known from the Pythagoric commentaries. He was also accustomed to pour forth sentences resembling Oracles to his familiars in a symbolical manner, and which in the greatest brevity of words contained the most abundant and multifarious meaning, like the Pythian Apollo through certain oracles, or like nature herself through seeds small in bulk, the former exhibiting conceptions, and the latter effects, innumerable in multitude, and difficult to be understood. Of this kind is the sentence, The beginning is the half of the whole , which is an apothegm of Pythagoras himself.

But not only in the present hemistich, but in others of a similar nature, the most divine Pythagoras has concealed the sparks of truth; depositing as in a treasury for those who are capable of being enkindled by them, and with a certain brevity of diction, an extension of theory most ample and difficult to be comprehended, as in the following hemistich: