(546) the laws which regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and they will bring children into the world when they ought not. Now that which is of divine birth has a period which is contained in a perfect number, 1 but the period of human birth is comprehended in a number in which first increments by involution and evolution [ or squared and cubed] obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. 2 The base of these (3) with a third added (4) when combined with five (20) and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great (400 = 4 × 100), 3 and the other a figure having one side equal to the former, but oblong, 4 consisting of a hundred numbers squared upon rational diameters of a square (i.e. omitting fractions), the side of which is five (7 × 7 = 49 × 100 = 4900), each of them being less by one (than the perfect square which includes the fractions, sc. 50) or less by 5 two perfect squares of irrational diameters (of a square the side of which is five = 50 + 50 = 100); and a hundred cubes of three (27 × 100 = 2700 + 4900 + 400 = 8000). Now this number represents a geometrical figure which has control over the good and evil of births. For when your guardians are ignorant of the law of births, and unite bride and bridegroom out of season, the children will not be goodly or fortunate. And though only the best of them will be appointed by their predecessors, still they will be unworthy to hold their fathers’ places, and when they come into power as guardians, they will soon be found to fail in taking care of us, the Muses, first by under-valuing music; which neglect will soon extend to gymnastic; and hence the young men of your State will be less cultivated. In the succeeding generation rulers will be appointed who have lost the guardian power of testing the metal of your

(5) Such, then, is the style of the example in arithmetic. And let the testimony of geometry be the tabernacle that was constructed, and the ark that was fashioned, - constructed in most regular proportions, and through divine ideas, by the gift of understanding, which leads us from things of sense to intellectual objects, or rather from these to holy things, and to the holy of holies. For the squares of wood indicate that the square form, producing fight angles, pervades all, and points out security. And the length of the structure was three hundred cubits, and the breadth fifty, and the height thirty; and above, the ark ends in a cubit, narrowing to a cubit from the broad base like a pyramid, the symbol of those who are purified and tested by fire. And this geometrical proportion has a place, for the transport of those holy abodes, whose differences are indicated by the differences of the numbers set down below.

(4) On another principle, 120 is a triangular number, and consists of the equality of the number 64, [which consists of eight of the odd numbers beginning with unity], the addition of which (1, 3, 5, 7, 9, 11, 13, 15) in succession generate squares; and of the inequality of the number 56, consisting of seven of the even numbers beginning with 2 (2, 4, 6, 8, 10, 12, 14), which produce the numbers that are not squares Again, according to another way of indicating. the number 120 consists of four numbers - of one triangle, 15; of another, a square, 25; of a third, a pentagon, 35; and of a fourth, a hexagon, 45. The 5 is taken according to the same ratio in each mode. For in triangular numbers, from the unity 5 comes 15; and in squares, 25; and of those in succession, proportionally. Now 25, which is the number 5 from unity, is said to be the symbol of the Levitical tribe. And the number 35 depends also on the arithmetic, geometric, and harmonic scale of doubles - 6, 8, 9, 12; the addition of which makes 35. In these days, the Jews say that seven months' children are formed. And the number 45 depends on the scale of triples - 6, 9, 12, 18 - the addition of which makes 45; and similarly, in these days they say that nine months' children are formed.

(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.

(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.)

(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.

(530) other proportion. No, he replied, such an idea would be ridiculous. And will not a true astronomer have the same feeling when he looks at the movements of the stars? Will he not think that heaven and the things in heaven are framed by the Creator of them in the most perfect manner? But he will never imagine that the proportions of night and day, or of both to the month, or of the month to the year, or of the stars to these and to one another, and any other things that are material and visible can also be eternal and subject to no deviation—that would be absurd; and it is equally absurd to take so much pains in investigating their exact truth. I quite agree, though I never thought of this before. Then, I said, in astronomy, as in geometry, we should employ problems, and let the heavens alone if we would approach the subject in the right way and so make the natural gift of reason to be of any real use. That, he said, is a work infinitely beyond our present astronomers. Yes, I said; and there are many other things which must also have a similar extension given to them, if our legislation is to be of any value. But can you tell me of any other suitable study? No, he said, not without thinking. Motion, I said, has many forms, and not one only; two of

(36) The digits 1 and 2 are not considered numbers by the Pythagoreans, because they typify the two supermundane spheres. The Pythagorean numbers, therefore, begin with 3, the triangle, and 4, the square. These added to the 1 and the 2, produce the 10, the great number of all things, the archetype of the universe. The three worlds were called receptacles. The first was the receptacle of principles, the second was the receptacle of intelligences, and the third, or lowest, was the receptacle of quantities.

(7) As the geometrician, who endeavours To square the circle, and discovers not, By taking thought, the principle he wants, Even such was I at that new apparition; I wished to see how the image to the circle Conformed itself, and how it there finds place; But my own wings were not enough for this, Had it not been that then my mind there smote A flash of lightning, wherein came its wish. Here vigour failed the lofty fantasy: But now was turning my desire and will, Even as a wheel that equally is moved, The Love which moves the sun and the other stars.

(30) TABLE XII, Figures 12-19. Figure 12 is called A Mirror of Astrological Aspects. Below it is an astrologer's wheel. Figure 13 is similar to Figure 12. Figure 14 is a secret alchemical formula. The words around the circle read: Out of one in all is all. Figure 15 is an unsatisfactory attempt to show the comparative sizes of the suns and planets and their distances from each other. Figure 16 is the solar system with its internal and spiritual heavens. A B is the solar system; C is the sphere of fixed stars; D, E, F, G are the systems of the spiritual worlds; H is the throne of the living God; J, K, L, M, and N are the Great Beyond, unmeasurable.

(112) The tetrad--4--was esteemed by the Pythagoreans as the primogenial number, the root of all things, the fountain of Nature and the most perfect number. All tetrads are intellectual; they have an emergent order and encircle the world as the Empyreum passes through it. Why the Pythagoreans expressed God as a tetrad is explained in a sacred discourse ascribed to Pythagoras, wherein God is called the Number of Numbers. This is because the decad, or 10, is composed of 1, 2, 3, and 4. The number 4 is symbolic of God because it is symbolic of the first four numbers. Moreover, the tetrad is the center of the week, being halfway between 1 and 7. The tetrad is also the first geometric solid.

(510) but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion? Yes, he said, I know. And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on—the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind? That is true. And of this kind I spoke as the intelligible, although in the search after it the soul is compelled to use hypotheses; not ascending to a first principle, because she is unable to rise above the region of hypothesis, but employing the objects of which the shadows below are resemblances in their turn as images, they having in relation to the shadows and reflections of them a greater distinctness, and therefore a higher value.

(3) That ten is a sacred number, it is superfluous to say now. And if the tables that were written were the work of God, they will be found to exhibit physical creation. For by the "finger of God" is understood the power of God, by which the creation of heaven and earth is accomplished; of both of which the tables will be understood to be symbols. For the writing and handiwork of God put on the table is the creation of the world.

(525) And they appear to lead the mind towards truth? Yes, in a very remarkable manner. Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician. That is true. And our guardian is both warrior and philosopher? Certainly. Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavour to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being. That is excellent, he said. Yes, I said, and now having spoken of it, I must add how charming the science is! and in how many ways it conduces to our desired end, if pursued in the spirit of a philosopher, and not of a shopkeeper! How do you mean? I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know

(1) CONCERNING the secret significance of numbers there has been much speculation. Though many interesting discoveries have been made, it may be safely said that with the death of Pythagoras the great key to this science was lost. For nearly 2500 years philosophers of all nations have attempted to unravel the Pythagorean skein, but apparently none has been successful. Notwithstanding attempts made to obliterate all records of the teachings of Pythagoras, fragments have survived which give clues to some of the simpler parts of his philosophy. The major secrets were never committed to writing, but were communicated orally to a few chosen disciples. These apparently dated not divulge their secrets to the profane, the result being that when death sealed their lips the arcana died with diem.

(6) This child once undertook and completely succeeded in raising the number 8 progressively up to the sixteenth power—in naming the result, 281,474,976,710,656 he was absolutely correct in every figure. He could raise any given number progressively up to the 10th power, with so much speed that the person putting down the figures on paper would frequently request him to manifest less speed. He gave instantly the square root of 106,929, and the cube root of 268,336,125. He could give the prime factors of very large numbers, and could detect large prime numbers instantly. Once asked how many minutes there were in forty-eight years, and before the question could be written down he answered "25,228,800", adding "and the number of seconds in such period is 1,513,728,000." The child, when questioned concerning his ability to give such answers, and to solve such difficult problems, was unable to give such information. He could say that he did not know how the answer came into his mind, but it was evident from watching him that some actual process was under way in his mind, and that there was no question of mere trick of memory in his feats. Moreover, it is important to note that he was totally ignorant of even the common rules of arithmetic, and could not "figure" on slate or paper even the simplest sum in addition or multiplication. It is interesting to note the sequel to this case, i.e., the fact that when a few years later the child was sent to the common schools and was there instructed in the art of written arithmetic, his power began to vanish, and eventually it left him altogether, and he became no more than any other child of his age. It seemed as if some door of his soul had been closed, while before it had stood ajar.

(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.

(602) and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an effect upon us like magic. True. And the arts of measuring and numbering and weighing come to the rescue of the human understanding—there is the beauty of them—and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight? Most true. And this, surely, must be the work of the calculating and rational principle in the soul? To be sure. And when this principle measures and certifies that some things are equal, or that some are greater or less than others, there occurs an apparent contradiction? True. But were we not saying that such a contradiction is impossible—the same faculty cannot have contrary opinions at the same time about the same thing? Very true. Then that part of the soul which has an opinion contrary to measure is not the same with that which has an opinion in accordance with measure? True. And the better part of the soul is likely to be that which trusts to measure and calculation? Certainly. And that which is opposed to them is one of the inferior principles of the soul? No doubt. This was the conclusion at which I was seeking to arrive when I said that painting or drawing, and imitation in general, when doing their own proper work, are far removed from truth, and the companions and friends and associates of

(36) Figure 9 is the solar system. Around the central part are the words The Place of the Damned. Figure 10 shows the dot, or point of rest, surrounded by a triangle enclosing a circle containing the names of the twelve tribes of Israel. It represents completion of the process of regeneration and the consummation of the Great Work.

(337) How characteristic of Socrates! he replied, with a bitter laugh;—that’s your ironical style! Did I not foresee—have I not already told you, that whatever he was asked he would refuse to answer, and try irony or any other shuffle, in order that he might avoid answering? You are a philosopher, Thrasymachus, I replied, and well know that if you ask a person what numbers make up twelve, taking care to prohibit him whom you ask from answering twice six, or three times four, or six times two, or four times three, ‘for this sort of nonsense will not do for me,’—then obviously, if that is your way of putting the question, no one can answer you. But suppose that he were to retort, ‘Thrasymachus, what do you mean? If one of these numbers which you interdict be the true answer to the question, am I falsely to say some other number which is not the right one?—is that your meaning?’—How would you answer him? Just as if the two cases were at all alike! he said. Why should they not be? I replied; and even if they are not, but only appear to be so to the person who is asked, ought he not to say what he thinks, whether you and I forbid him or not? I presume then that you are going to make one of the interdicted answers? I dare say that I may, notwithstanding the danger, if upon reflection I approve of any of them.