(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

(10) All higher numbers can be reduced to one of the original ten numerals, and the 10 itself to 1. Therefore, all groups of numbers resulting from the translation of names of deities into their numerical equivalents have a basis in one of the first ten numbers. By this system, in which the digits are added together, 666 becomes 6+6+6 or 18, and this, in turn, becomes 1+8 or 9. According to Revelation, 144,000 are to be saved. This number becomes 1+4+4+0+0+0, which equals 9, thus proving that both the Beast of Babylon and the number of the saved refer to man himself, whose symbol is the number 9. This system can be used successfully with both Greek and Hebrew letter values.

(74) Number is the term applied to all numerals and their combinations. (A strict interpretation of the term number by certain of the Pythagoreans excludes 1 and 2.) Pythagoras defines number to be the extension and energy of the spermatic reasons contained in the monad. The followers of Hippasus declared number to be the first pattern used by the Demiurgus in the formation of the universe.

(5) Another marked case is that of Zerah Colburn, the mathematical prodigy, whose feats attracted the attention of the scientific world during the last century. In this case, the child under eight years of age, without any previous knowledge of even the common rules of arithmetic, or even of the use and powers of the Arabic numerals, solved a great variety of arithmetical problems by a simple operation of the mind, and without the use of any visible symbols or contrivances. He could answer readily a question involving the statement of the exact number of minutes or seconds in any given period of time. He could also state with equal facility the exact product of the multiplication of any number containing two, three, or four figures by another number consisting of a like number of figures. He could state almost instantly all the factors composing a number of six or seven places of figures. He could likewise determine instantly questions concerning the extraction of the square and cube roots of any number proposed, and likewise whether it was a prime number incapable of division by any other number, for which there is no known general rule among mathematicians. Asked such questions in the midst of his ordinary childish play, he would answer them almost instantly and then proceed with his play.

(18) 3. The ten numbers (Sephiroth) out of Nothing are analogous to the ten fingers and the ten toes: five over against five. In the center between them is the covenant with the Only One God. In the spiritual world it is the covenant of the voice (the Word), and in the corporeal world the circumcision of the flesh (the rite of Abraham).

(83) 4. The numerical cipher. Many cryptograms have been produced in which numbers in various sequences are substituted for letters, words, or even complete thoughts. The reading of numerical ciphers usually depends upon the possession of specially arranged tables of correspondences. The numerical cryptograms of the Old Testament are so complicated that only a few scholars versed in rabbinical lore have ever sought to unravel their mysteries. In his Œdipus Ægyptiacus, Athanasius Kircher describes several Arabian Qabbalistic theorems, and a great part of the Pythagorean mystery was concealed in a secret method in vogue among Greek mystics of substituting letters for numbers.

(525) how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply 4 , taking care that one shall continue one and not become lost in fractions. That is very true. Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer? They would answer, as I should conceive, that they were speaking of those numbers which can only be realized in thought. Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth? Yes; that is a marked characteristic of it. And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been. Very true, he said.

(16) But Lot we saved ; for God remembered Abraham, and sent him out from the midst of the overthrow.

(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.

(3) The ten numbers formed from nothing are the Decad: these are seen in the fingers of the hands, five on one, five on the other, and over them is the Covenant by voice spiritual, and the rite of Circumcision, corporeal (as of Abraham).

(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.

(16) But here we may be questioned about these numbers which we describe as the primal and authentic:

"Where do you place these numbers, in what genus among Beings? To everyone they seem to come under Quantity and you have certainly brought Quantity in, where you say that discrete Quantity equally with the continuous holds place among Beings; but you go on to say that there are the numbers belonging to the Firsts and then talk of other numbers quite distinct, those of reckoning; tell us how you arrange all this, for there is difficulty here. And then, the unity in sense-things- is that a quantity or is quantity here just so many units brought together, the unity being the starting-point of quantity but not quantity itself? And, if the starting-point, is it a kindred thing or of another genus? All this you owe it to us to make clear."

Be it so; we begin by pointing out a distinction:

You take one thing with another- for we must first deal with objects of sense- a dog and a man, or two men; or you take a group and affirm ten, a decad of men: in this case the number affirmed is not a Reality, even as Reality goes in the sphere of sense, but is purely Quantity: similarly when you resolve into units, breaking up the decad, those units are your principle of Quantity since the single individual is not a unity absolute.

But the case is different when you consider one man in himself and affirm a certain number, duality, for example, in that he is at once living and reasoning.

By this analysis and totalling, you get quantity; but there are two objects under consideration and each of these is one; each of the unities contributes to the complete being and the oneness is inherent in each; this is another kind of number; number essential; even the duality so formed is no posterior; it does not signify a quantity apart from the thing but the quantity in the essence which holds the thing together. The number here is no mere result of your detailing; the things exist of themselves and are not brought together by your reckoning, but what has it to do with essential reality that you count one man in with another? There is here no resultant unity such as that of a choir- the decad is real only to you who count the ten; in the ten of your reckoning there cannot be a decad without a unitary basis; it is you that make the ten by your counting, by fixing that tenness down to quantity; in choir and army there is something more than that, something not of your placing.

But how do you come to have a number to place?

The Number inherent apart from any enumeration has its own manner of being, but the other, that resulting upon the appearance of an external to be appraised by the Number within yourself, is either an Act of these inherent numbers or an Act in accordance with them; in counting we produce number and so bring quantity into being just as in walking we bring a certain movement into being.

But what of that "Number within us having its own manner of being"?

It is the Number of our essence. "Our essence" we read "partakes of Number and harmony and, also, is Number and harmony." "Neither body nor magnitude," someone says: soul, then, is Number since it is essence. The number belonging to body is an essence of the order of body; the number belonging to soul constitutes the essences of souls.

In the Intellectuals, all, if the Absolute Living-Form, there is a multiple- a triad, let us say- that Triad of the Living-Form is of the nature of essence: and the Triad prior to any living thing, Triad in the realm of Being, is a principle of essence.

When you enumerate two things- say, animal and beauty- each of these remains one thing; the number is your production; it lay within yourself; it is you that elaborate quantity, here the dyad. But when you declare virtue to be a Tetrad, you are affirming a Tetrad which does actually exist; the parts, so to speak, make one thing; you are taking as the object of your act a Unity- Tetrad to which you accommodate the Tetrad within yourself.

(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.

(9) The second example is the mysterious Gnostic pantheos Abraxas. For this name the Greek table is used. Abraxas in Greek is Ἀβραξας. Α = 1, β = 2, ρ = 100, α = 1, ξ =60, α = 1, ς = 200, the sum being 365, the number of days in the year. This number furnishes the key to the mystery of Abraxas, who is symbolic of the 365 Æons, or Spirits of the Days, gathered together in one composite personality. Abraxas is symbolic of five creatures, and as the circle of the year actually consists of 360 degrees, each of the emanating deities is one-fifth of this power, or 72, one of the most sacred numbers in the Old Testament of the Jews and in their Qabbalistic system. This same method is used in finding the numerical value of the names of the gods and goddesses of the Greeks and Jews.

(12) There is also a system of calculation in vogue for the English language, but its accuracy is a matter of legitimate dispute. It is comparatively modern and has no relationship either to the Hebrew Qabbalistic system or to the Greek procedure. The claim made by some that it is Pythagorean is not supported by any tangible evidence, and there are many reasons why such a contention is untenable. The fact that Pythagoras used 10 as the basis of calculation, while this system uses 9--an imperfect number--is in itself almost conclusive. Furthermore, the arrangement of the Greek and Hebrew letters does not agree closely enough with the English to permit the application of the number sequences of one language to the number sequences of the others. Further experimentation with

(4) So he called every one of his lord's debtors unto him, and said unto the first, How much owest thou unto my lord? And he said, A hundred measures of oil. And he said unto him, Take thy bill, and sit down quickly, and write fifty.

(99) "Perfect numbers, therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be. And evil indeed is opposed to evil, but both are opposed to one good. Good, however, is never opposed to good, but to two evils at one and the same time. Thus timidity is opposed to audacity, to both [of] which the want of true courage is common; but both timidity and audacity are opposed to fortitude. Craft also is opposed to fatuity, to both [of] which the want of intellect is common; and both these are opposed to prudence. Thus, too, profusion is opposed to avarice, to both [of] which illiberality is common; and both these are opposed to liberality. And in a similar manner in the other virtues; by all [of] which it is evident that perfect numbers have a great similitude to the virtues. But they also resemble the virtues on another account; for they are rarely found, as being few, and they are generated in a very constant order. On the contrary, an infinite multitude of superabundant and diminished numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite."

(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.)

(134) The ennead--9--was the first square of an odd number (3x3). It was associated with failure and shortcoming because it fell short of the perfect number 10 by one. It was called the called the number of man, because of the nine months of his embryonic life. Among its keywords are ocean and horizon, because to the ancients these were boundless. The ennead is the limitless number because there is nothing beyond it but the infinite 10. It was called boundary and limitation, because it gathered all numbers within itself. It was called the sphere of the air, because it surrounded the numbers as air surrounds the earth, Among the gods and goddesses who partook in greater or less degree of its nature were Prometheus, Vulcan, Juno, the sister and wife of Jupiter, Pæan, and Aglaia, Tritogenia, Curetes, Proserpine, Hyperion, and Terpsichore (a Muse).

(24) Therefore our aeon came to be as the type of Immortal Man. Time came to be as the type of First Begetter, his son. The year came to be as the type of Savior. The twelve months came to be as the type of the twelve powers. The three hundred and sixty days of the year came to be as the three hundred and sixty powers who appeared from Savior. Their hours and moments came to be as the type of the angels who came from them (the powers), who are without number.