(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:

(15) The same holds also of astronomy. For treating of the description of the celestial objects, about the form of the universe, and the revolution of the heavens, and the motion of the stars, leading the soul nearer to the creative power, it teaches to quickness in perceiving the seasons of the year, the changes of the air, and the appearance of the stars; since also navigation and husbandry derive from this much benefit, as architecture and building from geometry. This branch of learning, too, makes the soul in the highest degree observant, capable of perceiving the true and detecting the false, of discovering correspondences and proportions, so as to hunt out for similarity in things dissimilar; and conducts us to the discovery of length without breadth, and superficial extent without thickness, and an indivisible point, and transports to intellectual objects from those of sense.

(1) It is also said, that Pythagoras was the first who called himself a philosopher; this not being a new name, but previously instructing us in a useful manner in a thing appropriate to the name. For he said that the entrance of men into the present life, resembled the progression of a crowd to some public spectacle. For there men of every description assemble with different views; one hastening to sell his wares for the sake of money and gain; but another that he may acquire renown by exhibiting the strength of his body; and there is also a third class of men, and those the most liberal, who assemble for the sake of surveying the places, the beautiful works of art, the specimens of valor, and the literary productions which are usually exhibited on such occasions.

Thus also in the present life, men of all-various pursuits are collected together in one and the same place. For some are influenced by the desire of riches and luxury; others by the love of power and dominion; and others are possessed with an insane ambition for glory. But the most pure and unadulterated character, is that of the man who gives himself to the contemplation of the most beautiful things, and whom it is proper to call a philosopher. He adds, that the survey of all heaven, and of the stars that revolve in it, is indeed beautiful, when the order of them is considered. For they derive this beauty and order by the participation of the first and the intelligible essence.

But that first essence is the nature of number and reasons [i. e. productive principles,] which pervades through all things, and according to which all these [celestial bodies] are elegantly arranged, and fitly adorned. And wisdom indeed, truly so called, is a certain science which is conversant with the first beautiful objects, and these divine, undecaying, and possessing an invariable sameness of subsistence; by the participation of which other things also may be called beautiful. But philosophy is the appetition of a thing of this kind. The attention therefore to erudition is likewise beautiful, which Pythagoras extended, in order to effect the correction of mankind.

(18) The study of geometry, music, and astronomy was considered essential to a rational understanding of God, man, or Nature, and no one could accompany Pythagoras as a disciple who was not thoroughly familiar with these sciences. Many came seeking admission to his school. Each applicant was tested on these three subjects, and if found ignorant, was summarily dismissed.

(4) For some they have borrowed, and others they have misunderstood. And in the case of others, what they have spoken, in consequence of being moved, they have not yet perfectly worked out; and others by human conjecture and reasoning, in which also they stumble. And they think that they have hit the truth perfectly; but as we understand them, only partially. They know, then, nothing more than this world. And it is just like geometry, which treats of measures and magnitudes and forms, by delineation on plane-surfaces; and just as painting appears to take in the whole field of view in the scenes represented. But it gives a false description of the view, according to the rules of the art, employing the signs that result from the incidents of the lines of vision. By this means, the higher and lower points in the view, and those between, are preserved; and some objects seem to appear in the foreground, and others in the background, and others to appear in some other way, on the smooth and level surface. So also the philosophers copy the truth, after the manner of painting. And always in the case of each one of them, their self-love is the cause of all their mistakes.

(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.

(1) [Trismegistus] ’Tis in this way, Asclepius;—by mixing it, by means of subtle expositions, with divers sciences not easy to be grasped,—such as arithmetic, and music, and geometry. But Pure Philosophy, which doth depend on godly piety alone, should only so far occupy itself with other arts, that it may [know how to] appreciate the working out in numbers of the fore-appointed stations of the stars when they return, and of the course of their procession. Let her, moreover, know how to appreciate the Earth’s dimensions, its qualities and quantities, the Water’s depths, the strength of Fire, and the effects and nature of all these. [And so] let her give worship and give praise unto the Art and Mind of God.

(58) An interesting application of the Pythagorean doctrine of geometric solids as expounded by Plato is found in The Canon. "Nearly all the old philosophers," says its anonymous author, "devised an harmonic theory with respect to the universe, and the practice continued till the old mode of philosophizing died out. Kepler (1596), in order to demonstrate the Platonic doctrine, that the universe was formed of the five regular solids, proposed the following rule. 'The earth is a circle, the measurer of all. Round it describe a dodecahedron; the circle inclosing this will be Mars. Round Mars describe a tetrahedron; the sphere inclosing this will be Jupiter. Describe a cube round Jupiter; the sphere containing this will be Saturn. Now inscribe in the earth an icosahedron; the circle inscribed in it will be Venus. Inscribe an octahedron in Venus; the circle inscribed in it will be Mercury' (Mysterium Cosmographicum, 1596). This rule cannot be taken seriously as a real statement of the proportions of the cosmos, fox it bears no real resemblance to the ratios published by Copernicus in the beginning of the sixteenth century. Yet Kepler was very proud of his formula, and said he valued it more than the Electorate of Saxony. It was also approved by those two eminent authorities, Tycho and Galileo, who evidently understood it. Kepler himself never gives the least hint of how his precious rule is to be interpreted." Platonic astronomy was not concerned with the material constitution or arrangement of the heavenly bodies, but considered the stars and planers primarily as focal points of Divine intelligence. Physical astronomy was regarded as the science of "shadows," philosophical astronomy the science of "realities."

(4) There was, however, a certain person named Hippomedon, an Ægean, a Pythagorean and one of the Acusmatici, who asserted that Pythagoras gave the reasons and demonstrations of all these precepts, but that in consequence of their being delivered to many, and these such as were of a more sluggish genius, the demonstrations were taken away, but the problems themselves were left. Those however of the Pythagoreans that are called Mathematici , acknowledge that these reasons and demonstrations were added by Pythagoras, and they say still more than this, and contend that their assertions are true, but affirm that the following circumstance was the cause of the dissimilitude. Pythagoras, say they, came from Ionia and Samos, during the tyranny of Polycrates, Italy being then in a florishing condition; and the first men in the city became his associates.

But, to the more elderly of these, and who were not at leisure [for philosophy], in consequence of being occupied by political affairs, the discourse of Pythagoras was not accompanied with a reasoning process, because it would have been difficult for them to apprehend his meaning through disciplines and demonstrations; and he conceived they would nevertheless be benefited by knowing what ought to be done, though they were destitute of the knowledge of the why : just as those who are under the care of physicians, obtain their health, though they do not hear the reason of every thing which is to be done to them. But with the younger part of his associates, and who were able both to act and learn,—with these he conversed through demonstration and disciplines.

These therefore are the assertions of the Mathematici, but the former, of the Acusmatici. With respect to Hippasus however especially, they assert that he was one of the Pythagoreans, but that in consequence of having divulged and described the method of forming a sphere from twelve pentagons, he perished in the sea, as an impious person, but obtained the renown of having made the discovery. In reality, however, this as well as every thing else pertaining to geometry, was the invention of that man ; for thus without mentioning his name, they denominate Pythagoras. But the Pythagoreans say, that geometry was divulged from the following circumstance: A certain Pythagorean happened to lose the wealth which he possessed; and in consequence of this misfortune, he was permitted to enrich himself from geometry.

But geometry was called by Pythagoras Historia . And thus much concerning the difference of each mode of philosophising, and the classes of the auditors of Pythagoras. For those who heard him either within or without the veil, and those who heard him accompanied with seeing, or without seeing him, and who are divided into interior and exterior auditors, were no other than these. And it is requisite to arrange under these, the political, economic and legislative Pythagoreans.

(69) The Pythagoreans declared arithmetic to be the mother of the mathematical sciences. This is proved by the fact that geometry, music, and astronomy are dependent upon it but it is not dependent upon them. Thus, geometry may be removed but arithmetic will remain; but if arithmetic be removed, geometry is eliminated. In the same manner music depends upon arithmetic, but the elimination of music affects arithmetic only by limiting one of its expressions. The Pythagoreans also demonstrated arithmetic to be prior to astronomy, for the latter is dependent upon both geometry and music. The size, form, and motion of the celestial bodies is determined by the use of geometry; their harmony and rhythm by the use of music. If astronomy be removed, neither geometry nor music is injured; but if geometry and music be eliminated, astronomy is destroyed. The priority of both geometry and music to astronomy is therefore established. Arithmetic, however, is prior to all; it is primary and fundamental.

(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.

(84) Far-sighted were the initiates of antiquity. They realized that nations come and go, that empires rise and fall, and that the golden ages of art, science, and idealism are succeeded by the dark ages of superstition. With the needs of posterity foremost in mind, the sages of old went to inconceivable extremes to make certain that their knowledge should be preserved. They engraved it upon the face of mountains and concealed it within the measurements of colossal images, each of which was a geometric marvel. Their knowledge of chemistry and mathematics they hid within mythologies which the ignorant would perpetuate, or in the spans and arches of their temples which time has not entirely obliterated. They wrote in characters that neither the vandalism of men nor the ruthlessness of the elements could completely efface, Today men gaze with awe and reverence upon the mighty Memnons standing alone on the sands of Egypt, or upon the strange terraced pyramids of Palanque. Mute testimonies these are of the lost arts and sciences of antiquity; and concealed this wisdom must remain until this race has learned to read the universal language--SYMBOLISM.

(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:

(1) If, however, it be necessary, dismissing these particulars, to speak what appears to me to be the truth, you do not rightly infer “ that a knowledge of this mathematical science cannot be obtained, because there is much dissonance concerning it, or because Chæremon, or some other, has written against it .” For if this reason were admitted, all things will be incomprehensible. For all sciences have ten thousand controvertists, and the doubts with which they are attended are innumerable. As, therefore, we are accustomed to say in opposition to the contentious, that contraries in things that are true are naturally discordant, and that it is not falsities alone that are hostile to each other; thus, also, we say respecting this mathematical science, that it is indeed true; but that those who wander from the scope of it, being ignorant of the truth, contradict it. This, however happens not in this science alone, but likewise in all the sciences, which are imparted by the Gods to men.

(11) Now as to the arts and crafts and their productions:

The imitative arts- painting, sculpture, dancing, pantomimic gesturing- are, largely, earth-based; on an earthly base; they follow models found in sense, since they copy forms and movements and reproduce seen symmetries; they cannot therefore be referred to that higher sphere except indirectly, through the Reason-Principle in humanity.

On the other hand any skill which, beginning with the observation of the symmetry of living things, grows to the symmetry of all life, will be a portion of the Power There which observes and meditates the symmetry reigning among all beings in the Intellectual Kosmos. Thus all music- since its thought is upon melody and rhythm- must be the earthly representation of the music there is in the rhythm of the Ideal Realm.

The crafts, such as building and carpentry which give us Matter in wrought forms, may be said, in that they draw on pattern, to take their principles from that realm and from the thinking There: but in that they bring these down into contact with the sense-order, they are not wholly in the Intellectual: they are founded in man. So agriculture, dealing with material growths: so medicine watching over physical health; so the art which aims at corporeal strength and well-being: power and well-being mean something else There, the fearlessness and self-sufficing quality of all that lives.

Oratory and generalship, administration and sovereignty- under any forms in which their activities are associated with Good and when they look to that- possess something derived thence and building up their knowledge from the knowledge There.

Geometry, the science of the Intellectual entities, holds place There: so, too, philosophy, whose high concern is Being.

For the arts and products of art, these observations may suffice.

(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.

(1) Since it is usual with all men of sound understandings, to call on divinity, when entering on any philosophic discussion, it is certainly much more appropriate to do this in the consideration of that philosophy which justly receives its denomination from the divine Pythagoras. For as it derives its origin from the Gods, it cannot be apprehended without their inspiring aid. To which we may also add, that the beauty and magnitude of it so greatly surpasses human power, that it is impossible to survey it by a sudden view; but then alone can any one gradually collect some portion of this philosophy, when, the Gods being his leaders, he quietly approaches to it. On all these accounts, therefore, having invoked the Gods as our leaders, and converting both ourselves and our discussion to them, we shall acquiesce in whatever they may command us to do.

We shall not, however, make any apology for this sect having been neglected for a long time, nor for its being concealed by foreign disciplines, and certain arcane symbols, nor for having been obscured by false and spurious writings, nor for many other such-like difficulties by which it has been impeded. For the will of the Gods is sufficient for us, in conjunction with which it is possible to sustain things still more arduous than these. But after the Gods, we shall unite ourselves as to a leader, to the prince and father of this divine philosophy; of whose origin and country we must rise a little higher in our investigation.

(64) Pythagoras saith: How marvellous is the diversity of the Philosophers in those things which they formerly asserted, and in their coming. together {or agreement], in respect of this small and most common-thing, wherein the precious thing is concealed! And if the vulgar knew, O all ye investigators of this art, the same small and vile thing, they would deem it a lie! Yet, if they knew its efficacy, they would not vilify it, but God hath concealed this from the crowd* lest the world should be devastated.

(10) Though I have not studied nor learned their arts, neither do I know how to go about to measure their circles; I take no great care about that. However, they will have so much to learn from hence, that many will not comprehend the ground thereof all the days of their lives.

(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: