(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.

(73) The monad signifies (a) the all-including ONE. The Pythagoreans called the monad the "noble number, Sire of Gods and men." The monad also signifies (b) the sum of any combination of numbers considered as a whole. Thus, the universe is considered as a monad, but the individual parts of the universe (such as the planets and elements) are monads in relation to the parts of which they themselves are composed, though they, in turn, are parts of the greater monad formed of their sum. The monad may also be likened (c) to the seed of a tree which, when it has grown, has many branches (the numbers). In other words, the numbers are to the monad what the branches of the tree are to the seed of the tree. From the study of the mysterious Pythagorean monad, Leibnitz evolved his magnificent theory of the world atoms--a theory in perfect accord with the ancient teachings of the Mysteries, for Leibnitz himself was an initiate of a secret school. By some Pythagoreans the monad is also considered (d) synonymous with the one.

(524) Most true. This was what I meant when I spoke of impressions which invited the intellect, or the reverse—those which are simultaneous with opposite impressions, invite thought; those which are not simultaneous do not. I understand, he said, and agree with you. And to which class do unity and number belong? I do not know, he replied. Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks ‘What is absolute unity?’ This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being. And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude? Yes, I said; and this being true of one must be equally true of all number? Certainly. And all arithmetic and calculation have to do with number? Yes.

(101) Monad--1--is so called because it remains always in the same condition--that is, separate from multitude. Its attributes are as follows: It is called mind, because the mind is stable and has preeminence; hermaphrodism, because it is both male and female; odd and even, for being added to the even it makes odd, and to the odd, even; God, because it is the beginning and end of all, but itself has neither beginning nor end; good, for such is the nature of God; the receptacle of matter, because it produces the duad, which is essentially material.

(6) The self-existent Super-goodness then, as projecting the first gift of self-existent being, is celebrated by the elder and first of the participations; and being itself is from It, and in It; as also the sources of things being, and all the things that be, and the things howsoever sustained by being, and that irresistibly, and comprehensively and uniformly. For even in a monad, every number preexists in the form of a unit, and the monad holds every number in itself singly. And every number is united in the monad, but so far as it advances from the monad, so far it is distributed and multiplied. And in a centre, all the lines of the circle coexist within one union, and the point holds all the straight lines in itself, uniformly united, both to each other, and to the one source from which they proceeded, and in the centre itself they are completely united; but standing slightly distant from it, they are slightly separated; but when more apart, more so. And in one word, the nearer they are to the centre, the more they are united to it and to each other? and the more they stand apart from it, the more they stand apart from each other.

(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

(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.

(9) Again the Barbarian philosophy knows the world of thought and the world of sense - the former archetypal, and the latter the image of that which is called the model; and assigns the former to the Monad, as being perceived by the mind, and the world of sense to the number six. For six is called by the Pythagoreans marriage, as being the genital number; and he places in the Monad the invisible heaven and the holy earth, and intellectual light. For "in the beginning," it is said, "God made the heaven and the earth; and the earth was invisible." And it is added, "And God said, Let there be light; and there was light." And in the material cosmogony He creates a solid heaven (and what is solid is capable of being perceived by sense), and a visible earth, and a light that is seen. Does not Plato hence appear to have left the ideas of living creatures in the intellectual world, and to make intellectual objects into sensible species according to their genera? Rightly then Moses says, that the body which Plato calls "the earthly tabernacle" was formed of the ground, but that the rational soul was breathed by God into man's face. For there, they say, the ruling faculty is situated; interpreting the access by the senses into the first man as the addition of the soul.

(9) the Pure intelligence so called because it purifies the Numerations, it proves and corrects the designing of their representation, and disposes their unity with which they are combined without diminution or division.

(76) There are two orders of number: odd and even. Because unity, or 1, always remains indivisible, the odd number cannot be divided equally. Thus, 9 is 4+1+4, the unity in the center being indivisible. Furthermore, if any odd number be divided into two parts, one part will always be odd and the other even. Thus, 9 may be 5+4, 3+6, 7+2, or 8+1. The Pythagoreans considered the odd number--of which the monad was the prototype--to be definite and masculine. They were not all agreed, however, as to the nature of unity, or 1. Some declared it to be positive, because if added to an even (negative) number, it produces an odd (positive) number. Others demonstrated that if unity be added to an odd number, the latter becomes even, thereby making the masculine to be feminine. Unity, or 1, therefore, was considered an androgynous number, partaking of both the masculine and the feminine attributes; consequently both odd and even. For this reason the Pythagoreans called it evenly-odd. It was customary for the Pythagoreans to offer sacrifices of an uneven number of objects to the superior gods, while to the goddesses and subterranean spirits an even number was offered.

(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."

(2) But One, because He is uniquely all, as beseems an excess of unique Oneness, and is Cause of all without departing from the One. For there is no single existing being, which does not participate in the one, but as every number participates in an unit, and one dual and one decade is spoken of, and one half, and one third and tenth, so everything, and part of everything participates in the one, and by the fact that the One is, all existing things are. And the Cause of all is not One, as one of many, but before every one and multitude, and determinative of every one and multitude. For there is no multitude which does not partake in some way or other of the one. Yea, that which is many by parts, is one in the whole; and the many by the accidents, is one by the subject; and the many by the number or the powers, is one by the species, and the many by the species, is one by the genus; and the many by the progressions, is one by the source. And there is no single thing which does not participate in some way in the one, which uniformly pre-held in the uniqueness throughout all, all and whole, all, even the things opposed. And indeed, without the one there will not be a multitude, but without the multitude there will be the one, even as the unit previous to every multiplied number; and, if any one should suppose, that all things are united to all, the All will be one in the whole.

(54) German philosophy had its inception with Gottfried Wilhelm von Leibnitz, whose theories are permeated with the qualities of optimism and idealism. Leibnitz's criteria of sufficient reason revealed to him the insufficiency of Descartes' theory of extension, and he therefore concluded that substance itself contained an inherent power in the form of an incalculable number of separate and all-sufficient units. Matter reduced to its ultimate particles ceases to exist as a substantial body, being resolved into a mass of immaterial ideas or metaphysical units of power, to which Leibnitz applied the term monad. Thus the universe is composed of an infinite number of separate monadic entities unfolding spontaneously through the objectification of innate active qualities. All things are conceived as consisting of single monads of varying magnitudes or of aggregations of these bodies, which may exist as physical, emotional, mental, or spiritual substances. God is the first and greatest Monad; the spirit of man is an awakened monad in contradistinction to the lower kingdoms whose governing monadic powers are in a semi-dormant state.

(8) And from the same Cause of all, are the higher and lower intellectual essences of the godlike angels; and those of the souls; and the natures of the whole Cosmos; all things whatsoever said to be either in others, or by reflection. Yea, even the all holy and most honoured Powers veritably being, and established, as it were, in the vestibule of the superessential Triad, are from It, and in It; and have the being and the godlike being; and after them, as regards Angels, the subordinate, subordinately, and the remotest, most remotely, but as regards ourselves, supermundanely. And the souls, and all the other beings, according to the same rule, have their being, and their well-being; and are, and are well; by having from the Pre-existing their being and their well-being. And in It are both being and well-being; and from It, beginning; and in It, guarded; and to It, terminated. And the prerogatives of being he distributes to the superior beings, which the Oracles call even eternal. But being itself never at any time fails all existing beings. And even self-existent being is from the Pre-existent, and of Him is being, and He is not of being;--and in Him is being, and He is not in being; and being possesses Him, and not He possesses being; and He is both age and beginning, and measure of being; being essentiating Source, and Middle and End, of pre-essence, and being and age and all things. And for this reason, by the Oracles, the veritably Pre-existing is represented under many forms, according to every conception of beings, and the "Was" and the "Is," and the "Will be," and the "Became," and the "Becomes," and the "Will become," are properly sung respecting Him. For all these, to those who think worthily of God, signify by every conception His being superessentially, and Cause in every way of things existing. For He is not this, but not that; nor is He in some way, but not in some other; but He is all things, as Cause of all, and containing and pre-holding in Himself all governments, all controls, of all existing things. And He is above all, as superessentially super-being before all. Wherefore, also, all things are predicated of Him and together, and He is none of them all; of every shape, of every kind, without form, without beauty, anticipating in Himself, beginnings and middles, and ends of things existing, irresistibly and preeminently; and shedding forth without flaw, (the light of) being to all, as beseems a One and super-united Cause. For, if our sun, at the same time that he is one and sheds a uniform light, renews the essences and qualities of sensible creatures, although they are many and various, and nourishes and guards, and perfects and distinguishes, and unites, and fosters, and makes to be productive, and increases, and transforms, and establishes, and makes to grow, and awakens, and gives life to all; and each of the whole, in a manner appropriate to itself, participates in the same and one sun; and the one sun anticipated in himself, uniformly, the causes of the many participants; much more with regard to the Cause of it and of all things, ought we to concede that It first presides over, as beseems One superessential Oneness, all the exemplars, of things existing; since He produces even essences, as beseems the egression from essence. But, we affirm that the exemplars are the methods in God, giving essence to things that be, and pre-existing uniformly, which theology calls predeterminations, and Divine and good wills, which define and produce things existing; according to which (predeterminations) the Superessential both predetermined and brought into existence everything that exists.

(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).

(23) "First Man has his unique mind, within, and thought - just as he is it (thought) - (and) considering, reflecting, rationality, power. All the attributes that exist are perfect and immortal. In respect to imperishableness, they are indeed equal. (But) in respect to power, they are different, like the difference between father and son and thought, and the thought and the remainder. As I said earlier, among the things that were created, the monad is first.

(1) As then in astronomy we have Abraham as an instance, so also in arithmetic we have the same Abraham. "For, hearing that Lot was taken captive, and having numbered his own servants, born in his house, 318 (tih)," he defeats a very great number of the enemy.

(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).

(60) The comic poet Epicharmus speaks in the Republic clearly of the Word in the following terms: "The life of men needs calculation and number alone, And we live by number and calculation, for these save mortals."

(10) All of them exist in the single one, as he clothes himself completely and by his single name he is never called. And in this unique way they are equally the single one and the Totalities. He is neither divided as a body, nor is he separated into the names which he has received, (so that) he is one thing in this way and another in another way. Also, neither does he change in [...], nor does he turn into the names which he thinks of, and become now this, now something else, this thing now being one thing and, at another time, something else, but rather he is wholly himself to the uttermost. He is each and every one of the Totalities forever at the same time. He is what all of them are. He brought the Father to the Totalities. He also is the Totalities, for he is the one who is knowledge for himself and he is each one of the properties. He has the powers and he is beyond all that which he knows, while seeing himself in himself completely and having a Son and form. Therefore, his powers and properties are innumerable and inaudible, because of the begetting by which he begets them. Innumerable and indivisible are the begettings of his words, and his commands and his Totalities. He knows them, which things he himself is, since they are in the single name, and are all speaking in it. And he brings (them) forth, in order that it might be discovered that they exist according to their individual properties in a unified way. And he did not reveal the multitude to the Totalities at once nor did he reveal his equality to those who had come forth from him.