(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.

(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

(2) Certain of the secret schools in the world today are perpetuations of the ancient Mysteries, and although it is quite possible that they may possess some of the original numerical formulæ, there is no evidence of it in the voluminous writings which have issued from these groups during the last five hundred years. These writings, while frequently discussing Pythagoras, show no indication of a more complete knowledge of his intricate doctrines than the post-Pythagorean Greek speculators had, who talked much, wrote little, knew less, and concealed their ignorance under a series of mysterious hints and promises. Here and there among the literary products of early writers are found enigmatic statements which they made no effort: to interpret. The following example is quoted from Plutarch:

(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.

(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) "Child, let us embrace with love. Be happy. Already the power, light, is coming to us from them. I see, I see ineffable depths. How shall I tell you, child? . . . How shall I tell you about the universe? I am mind, and I see another mind, one that moves the soul. I see the one that moves me from pure forgetfulness. You have given me power. I see myself. I want to speak. Fear seizes me. I have found the beginning of the power above all powers, without beginning. I see a spring bubbling with life. I have said, child, that I am mind. I have seen. Speech cannot reveal this. For all of the eighth, child, and the souls in it, and the angels, sing a song in silence. I, mind, understand."

(6) All his prayer and remembering were numerous powers according to that limit. For there is nothing barren in his thought.

(14) The purpose of this lesson is to impress upon the minds of our students the fact that, to all intents and purposes, the Universe and its laws, and its phenomena, are just as REAL, so far as Man is concerned, as they would be under the hypotheses of Materialism or Energism. Under any hypothesis the Universe in its outer aspect is changing, ever-flowing, and transitory--and therefore devoid of substantiality and reality. But (note the other pole of the truth) under the same hypotheses, we are compelled to ACT AND LIVE as if the fleeting things were real and substantial. With this difference, always, between the various hypotheses--that under the old views Mental Power was ignored as a Natural Force, while under Mentalism it becomes the Greatest Natural Force. And this one difference revolutionizes Life, to those who understand the Principle and its resulting laws and practice.

(13) Again it came to the third power, which is called ignorance.

(11) And the child began to understand the errors of the earth that all went astray after graven images and after uncleanness, and his father taught him 890 a.m. writing, and he was two weeks of years old, and he separated himself from his father, thathe might not worship idols with him.

(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

(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.

(522) Can we deny that a warrior should have a knowledge of arithmetic? Certainly he should, if he is to have the smallest understanding of military tactics, or indeed, I should rather say, if he is to be a man at all. I should like to know whether you have the same notion which I have of this study? What is your notion? It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being. Will you explain your meaning? he said. I will try, I said; and I wish you would share the enquiry with me, and say ‘yes’ or ‘no’ when I attempt to distinguish in my own mind what branches of knowledge have this attracting power, in order that we may have clearer proof that arithmetic is, as I suspect, one of them. Explain, he said. I mean to say that objects of sense are of two kinds; some of them do not invite thought because the sense is an adequate judge of them; while in the case of other objects sense is so untrustworthy that further enquiry is imperatively demanded. You are clearly referring, he said, to the manner in which the senses are imposed upon by distance, and by painting in light and shade. No, I said, that is not at all my meaning. Then what is your meaning? When speaking of uninviting objects, I mean those which

(113) Pythagoras maintained that the soul of man consists of a tetrad, the four powers of the soul being mind, science, opinion, and sense. The tetrad connects all beings, elements, numbers, and seasons; nor can anything be named which does not depend upon the tetractys. It is the Cause and Maker of all things, the intelligible God, Author of celestial and sensible good, Plutarch interprets this tetractys, which he said was also called the world, to be 36, consisting of the first four odd numbers added to the first four even numbers, thus:

(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.

(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.

(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.

(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.

(9) ’Tis mind that all things sees and hears;

What else exists is deaf and blind.

But the precept which is next to this in efficacy is that which exhorts to be beyond measure studious of purifying the intellect, and by various methods adapting it through mathematical orgies to receive something divinely beneficial, so as neither to fear a separation from body, nor, when led to incorporeal natures, to be forced to turn away the eyes, through their most refulgent splendor, nor to be converted to those passions which nail and fasten the soul to the body. And, in short, which urges the soul to be untamed by all those passions which are the progeny of the realms of generation, and which draw it to an inferior condition of being. For the exercise and ascent through all these, is the study of the most perfect fortitude. And such are the instances adduced by us of the fortitude of Pythagoras, and the Pythagoreans.

(2) Eternity, for instance, presents in an instant the future and the present, also the past of time. But truth, much more powerful than limitless duration, can collect its proper germs, though they have fallen on foreign soil. For we shall find that very many of the dogmas that are held by such sects as have not become utterly senseless, and are not cut out from the order of nature (by cutting off Christ, as the women of the fable dismembered the man), though appearing unlike one another, correspond in their origin and with the truth as a whole. For they coincide in one, either as a part, or a species, or a genus. For instance, though the highest note is different from the lowest note, yet both compose one harmony. And in numbers an even number differs from an odd number; but both suit in arithmetic; as also is the case with figure, the circle, and the triangle, and the square, and whatever figures differ from one another.