(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.

(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:

(527) Yes, that is what we assert. Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is in flat contradiction to the ordinary language of geometricians. How so? They have in view practice only, and are always speaking, in a narrow and ridiculous manner, of squaring and extending and applying and the like—they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science. Certainly, he said. Then must not a further admission be made? What admission? That the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient. That, he replied, may be readily allowed, and is true. Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Nothing will be more likely to have such an effect. Then nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. Moreover the science has indirect effects, which are not small. Of what kind? he said. There are the military advantages of which you spoke, I said; and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not. Yes indeed, he said, there is an infinite difference between them. Then shall we propose this as a second branch of knowledge which our youth will study? Let us do so, he replied.

(529) nothing of that sort is matter of science; his soul is looking downwards, not upwards, whether his way to knowledge is by water or by land, whether he floats, or only lies on his back. I acknowledge, he said, the justice of your rebuke. Still, I should like to ascertain how astronomy can be learned in any manner more conducive to that knowledge of which we are speaking? I will tell you, I said: The starry heaven which we behold is wrought upon a visible ground, and therefore, although the fairest and most perfect of visible things, must necessarily be deemed inferior far to the true motions of absolute swiftness and absolute slowness, which are relative to each other, and carry with them that which is contained in them, in the true number and in every true figure. Now, these are to be apprehended by reason and intelligence, but not by sight. True, he replied. The spangled heavens should be used as a pattern and with a view to that higher knowledge; their beauty is like the beauty of figures or pictures excellently wrought by the hand of Daedalus, or some other great artist, which we may chance to behold; any geometrician who saw them would appreciate the exquisiteness of their workmanship, but he would never dream of thinking that in them he could find the true equal or the true double, or the truth of any

(1) [Asclepius] And does the Cosmos have a species, O Thrice-greatest one?

[Trismegistus] Dost not thou see, Asclepius, that all has been explained to thee as though to one asleep? For what is Cosmos, or of what doth it consist, if not of all things born? This, then, you may assert of heaven, and earth, and elements. For though the other things possess more frequent change of species, [still even] heaven, [by its] becoming moist, or dry, or cold, or hot, or clear, or dull, [all] in one kind of heaven,—these [too] are frequent changes into species.

(3) A: Assuredly. H: Of what size, then, must be the space in which it's moved, and of what kind [must be] the nature [of that space]? Must it not be far vaster [than the cosmos], in order that it may be able to find room for its continued course, so that the moved may not be cramped for want of room and lose its motion? A: Something, Thrice-greatest one, it needs must be, immensely vast.

(6) If space is, therefore, to be thought, [it should] not, [then, be thought as] God, but space. If God is also to be thought, [He should] not [be conceived] as space, but as energy that can contain [all space]. Further, all that is moved is moved not in the moved but in the stable. And that which moves [another] is of course stationary, for 'tis impossible that it should move with it. A: How is it, then, that things down here, Thrice-greatest one, are moved with those that are [already] moved? For thou hast said the errant spheres were moved by the inerrant one. H: This is not, O Asclepius, a moving with, but one against; they are not moved with one another, but one against the other. It is this contrariety which turneth the resistance of their motion into rest. For that resistance is the rest of motion.

(7) Hence, too, the errant spheres, being moved contrarily to the inerrant one, are moved by one another by mutual contrariety, [and also] by the spable one through contrariety itself. And this can otherwise not be. The Bears up there , which neither set nor rise, think'st thou they rest or move? A: They move, Thrice-greatest one. H: And what their motion, my Asclepius? A: Motion that turns for ever round the same. H: But revolution - motion around same - is fixed by rest. For "round-the-same" doth stop "beyond-same". "Beyond-same" then, being stopped, if it be steadied in "round-same" - the contrary stands firm, being rendered ever stable by its contrariety.

(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

(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.

(24) "The sistrum is designed * * * to represent to us, that every thing must be kept in continual agitation, and never cease from motion; that they ought to be mused and well-shaken, whenever they begin to grow drowsy as it were, and to droop in their motion. For, say they, the sound of these sistra averts and drives away Typho; meaning hereby, that as corruption clogs and puts a stop to the regular course of nature; so generation, by the means of motion, loosens it again, and restores it to its former vigour. Now the outer surface of this instrument is of a convex figure, as within its circumference are contained those four chords or bars [only three shown], which make such a rattling when they are shaken--nor is this without its meaning; for that part of the universe which is subject to generation and corruption is contained within the sphere of the moon; and whatever motions or changes may happen therein, they are all effected by the different combinations of the four elementary bodies, fire, earth, water, and air--moreover, upon the upper part of the convex surface of the sistrum is carved the effigies of a cat with a human visage, as on the lower edge of it, under those moving chords, is engraved on the one side the face of Isis, and on the other that of Nephthys--by the faces symbolically representing generation and corruption (which, as has been already observed, is nothing but the motion and alteration of the four elements one amongst another),"

(527) And suppose we make astronomy the third—what do you say? I am strongly inclined to it, he said; the observation of the seasons and of months and years is as essential to the general as it is to the farmer or sailor. I am amused, I said, at your fear of the world, which makes you guard against the appearance of insisting upon useless studies; and I quite admit the difficulty of believing that in every man there is an eye of the soul which, when by other pursuits lost and dimmed, is by these purified and re-illumined; and is more precious far than ten thousand bodily eyes, for by it alone is truth seen. Now there are two classes of persons: one class of those who will agree with you and will take your words as a revelation; another class to whom they will be utterly unmeaning, and who will naturally deem them to be idle tales, for they see no sort of profit which is to be obtained from them. And therefore you had better decide at once with which of the two you are proposing to argue. You will very likely say with neither, and that your chief aim in carrying on the argument is your own improvement; at the same time you do not grudge to others any benefit which they may receive. I think that I should prefer to carry on the argument mainly on my own behalf. Then take a step backward, for we have gone wrong in the order of the sciences. What was the mistake? he said. After plane geometry, I said, we proceeded at once to

(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.

(12) At this point I have just recollected the following. In the end of the Timoeus he says: "You must necessarily assimilate that which perceives to that which is perceived, according to its original nature; and it is by so assimilating it that you attain to the end of the highest life proposed by the gods to men, for the present or the future time." For those have equal power with these. He, who seeks, will not stop till he find; and having found, he will wonder; and wondering, he will reign; and reigning, he will rest. And what? Were not also those expressions of Thales derived from these? The fact that God is glorified for ever, and that He is expressly called by us the Searcher of hearts, he interprets. For Thales being asked, What is the divinity? said, What has neither beginning nor end. And on another asking, "If a man could elude the knowledge of the Divine Being while doing aught?" said, "How could he who cannot do so while thinking?"

(21) The claim of Motion to be established as a genus will depend upon three conditions: first, that it cannot rightly be referred to any other genus; second, that nothing higher than itself can be predicated of it in respect of its essence; third, that by assuming differences it will produce species. These conditions satisfied, we may consider the nature of the genus to which we shall refer it.

Clearly it cannot be identified with either the Substance or the Quality of the things which possess it. It cannot, further, be consigned to Action, for Passivity also comprises a variety of motions; nor again to Passivity itself, because many motions are actions: on the contrary, actions and passions are to be referred to Motion.

Furthermore, it cannot lay claim to the category of Relation on the mere ground that it has an attributive and not a self-centred existence: on this ground, Quality too would find itself in that same category; for Quality is an attribute and contained in an external: and the same is true of Quantity.

If we are agreed that Quality and Quantity, though attributive, are real entities, and on the basis of this reality distinguishable as Quality and Quantity respectively: then, on the same principle, since Motion, though an attribute has a reality prior to its attribution, it is incumbent upon us to discover the intrinsic nature of this reality. We must never be content to regard as a relative something which exists prior to its attribution, but only that which is engendered by Relation and has no existence apart from the relation to which it owes its name: the double, strictly so called, takes birth and actuality in juxtaposition with a yard's length, and by this very process of being juxtaposed with a correlative acquires the name and exhibits the fact of being double.

What, then, is that entity, called Motion, which, though attributive, has an independent reality, which makes its attribution possible- the entity corresponding to Quality, Quantity and Substance?

But first, perhaps, we should make sure that there is nothing prior to Motion and predicated of it as its genus.

Change may be suggested as a prior. But, in the first place, either it is identical with Motion, or else, if change be claimed as a genus, it will stand distinct from the genera so far considered: secondly, Motion will evidently take rank as a species and have some other species opposed to it- becoming, say- which will be regarded as a change but not as a motion.

What, then, is the ground for denying that becoming is a motion? The fact, perhaps, that what comes to be does not yet exist, whereas Motion has no dealings with the non-existent. But, on that ground, becoming will not be a change either. If however it be alleged that becoming is merely a type of alteration or growth since it takes place when things alter and grow, the antecedents of becoming are being confused with becoming itself. Yet becoming, entailing as it does these antecedents, must necessarily be a distinct species; for the event and process of becoming cannot be identified with merely passive alteration, like turning hot or white: it is possible for the antecedents to take place without becoming as such being accomplished, except in so far as the actual alteration has "come to be"; where, however, an animal or a vegetal life is concerned, becoming takes place only upon its acquisition of a Form.

The contrary might be maintained: that change is more plausibly ranked as a species than is Motion, because change signifies merely the substitution of one thing for another, whereas Motion involves also the removal of a thing from the place to which it belongs, as is shown by locomotion. Even rejecting this distinction, we must accept as types of Motion knowledge and musical performance- in short, changes of condition: thus, alteration will come to be regarded as a species of Motion- namely, motion displacing.

(6) This uncertainty shared by all true students of the subject proves conclusively that it is unwise to make definite statements founded on the indefinite and fragmentary information available concerning the Pythagorean system of mathematical philosophy. The material which follows represents an effort to collect a few salient points from the scattered records preserved by disciples of Pythagoras and others who have since contacted his philosophy.

(71) Magnitude is divided into two parts--magnitude which is stationary and magnitude which is movable, the stationary pare having priority. Multitude is also divided into two parts, for it is related both to itself and to other things, the first relationship having priority. Pythagoras assigned the science of arithmetic to multitude related to itself, and the art of music to multitude related to other things. Geometry likewise was assigned to stationary magnitude, and spherics (used partly in the sense of astronomy) to movable magnitude. Both multitude and magnitude were circumscribed by the circumference of mind. The atomic theory has proved size to be the result of number, for a mass is made up of minute units though mistaken by the uninformed for a single simple substance.

(526) And indeed, you will not easily find a more difficult study, and not many as difficult. You will not. And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up. I agree. Let this then be made one of our subjects of education. And next, shall we enquire whether the kindred science also concerns us? You mean geometry? Exactly so. Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general is or is not a geometrician. Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates rather to the greater and more advanced part of geometry— whether that tends in any degree to make more easy the vision of the idea of good; and thither, as I was saying, all things tend which compel the soul to turn her gaze towards that place, where is the full perfection of being, which she ought, by all means, to behold. True, he said. Then if geometry compels us to view being, it concerns us; if becoming only, it does not concern us?

(8) Of this I'll give thee here on earth an instance, which the eye can see. Regard the animals down here - a man, for instance, swimming! The water moves, yet the resistance of his hands and feet give him stability, so that he is not borne along with it, nor sunk thereby. A: Thou hast, Thrice-greatest one, adduced a most clear instance. H: All motion, then, is caused in station and by station. The motion, therefore, of the cosmos (and of every other hylic to the cosmos, but by things interior [outward] to the exterior - such [things] as soul, or spirit, or some such other thing incorporeal. 'Tis not the body that doth move the living thing in it; nay, not even the whole [body of the universe a lesser] body e'en though there be no life in it.