The Pythagorean Sourcebook and Library

That's French for "the ancient system," as in the ancient system of feudal privileges and the exercise of autocratic power over the peasants. The ancien regime never goes away, like vampires and dinosaur bones they are always hidden in the earth, exercising a mysterious influence. It is not paranoia to believe that the elites scheme against the common man. Inform yourself about their schemes here.

Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 4:56 am


HIEROCLES was a Neoplatonic philosopher of Alexandria during the fifth century of the common era. His Commentaries on the Golden Verses of Pythagoras survives intact, and fragments remain of a treatise On Providence, Fate and Free Will. The ethical fragments assembled here are preserved by Stobaeus.

The only thing known about his life is an anecdote preserved by Suidas. In the same way that some Romans mistreated the early Christians, so too did later Christians abuse others who did not share their faith. The story preserved by Suidas demonstrates that Hierocles maintained an admirable sense of humor even under the most adverse conditions: upon arriving in Byzantium he seems to have offended certain Christians and was therefore whipped in the presence of a Christian magistrate. Taking some of his blood into the cup of his hand, Hierocles sprinkled the judge with it, quoting the lines from the Odyssey: "Cyclops, since human flesh is thy delight / Now drink this wine."

As the following fragments and his Commentaries aptly demonstrate, Hierocles was greatly gifted as a writer and was especially adept in the realm of ethical matters. For a study of his thought see Le Neo-Platonisme Alexandrin: Hierocles d'Alexandre, Leiden, E.J. Brill, 1987.


1. On How We Ought to Conduct Ourselves Towards the Gods

CONCERNING THE GODS we should assume that they are immutable and do not change their decrees; from the very beginning they never vary their conceptions of propriety, The immutability and firmness of the virtues we know, and reason suggests that it must transcendently be so with the Gods, and be the element which to their conceptions imparts a never-failing stability. Evidently no punishment which divinity thinks proper to inflict is likely to be remitted. For if the Gods changed their decisions, and omitted to punish someone whom they had designed to punish, the world could be neither beautifully nor justly governed; nor can we assign any probable reason for repentance on their part. Rashly, indeed, and without any reason, have poets written words such as the following:

Men bend the Gods, by incense and libation,
By gentle vows, and sacrifice and prayer,
When they transgress, and stray from what is right!
(Homer, Iliad, ix. 495-7)


Flexible are e'en the Gods themselves!
(ibid., verse 493)

Nor is this the only expression in poetry.

Nor must we omit to observe, that though the Gods are not the causes of evil, yet they connect certain persons with things of this kind, and surround those who deserve to be afflicted with corporeal and external hindrances; not through any malignity, or because they think it advisable that men should struggle with difficulties, but for the sake of punishment. For as in general pestilence and drought, rain storms, earthquakes and the like, are indeed for the most part produced by natural causes, yet they are sometimes caused by the Gods when the times are such that the multitude's iniquity needs to be punished publicly and in common; likewise the Gods sometimes afflict an individual with corporeal and external difficulties, in order to punish him and convert others to what is right.

The belief that the Gods are never the cause of any evil, it seems to me, contributes greatly to proper conduct towards the Gods. For evils proceed from vice alone, while the Gods are of themselves the causes of good, and of any advantage, though in the meantime we slight their beneficence, and surround ourselves with voluntary evils. That is why I agree with the poet who says,

--- that mortals blame the Gods

as if they were the causes of their evils!

---though not from fate,
But for their crimes they suffer woe!
(Homer, Odyssey, i. 32-34)

Many arguments prove that God is never in any way the cause of evil, but it will suffice to read [in the first book of the Republic] the words of Plato "that as it is not the nature of heat to refrigerate, so the beneficent cannot harm; but the contrary." Moreover, God being good, and from the beginning replete with every virtue, cannot harm nor cause evil to anyone; on the contrary, he imparts good to all willing to receive it, bestowing on us also such indifferent things as flow from nature, and which result in accordance with nature. But there is only one cause of evil.

2. On How We Ought to Conduct Ourselves Towards Our Country

AFTER SPEAKING OF THE GODS it is most reasonable, in the second place, to show how we should conduct ourselves towards our country. For God is my witness that our country is a sort of secondary divinity, and our first and greatest parent. That is why its name is, for good reason, patris, derived from pater, a father, but taking a feminine termination, to be as it were a mixture of father and mother. This also explains that our country should be honored equally with our parents, preferring it to either of them separately, and not even to it preferring both of our parents; preferring it besides to our wife, children and friends, and in short to all things, under the Gods.

He who would esteem one finger more than five would be considered stupid, inasmuch as it is reasonable to prefer five to one; the former despising the most desirable, while the latter among the five preserves also the one finger. Likewise, he who prefers to save himself rather than his country, in addition to acting unlawfully, desires an impossibility. On the contrary, he who to himself prefers his country is dear to divinity and reasons properly and irrefutably. Moreover it has been observed that though someone should not be a member of an organized society, remaining apart therefrom, yet it it proper that he should prefer the safety of society to his own; for the city's destruction would demonstrate that on its existence depended that of the individual citizen, just as the amputation of the hand involves the destruction of the finger as an integral part. We may therefore draw the general conclusion that general utility cannot be separated from private welfare, both at bottom being identical. For whatever is beneficial to the whole country is common to every single part, inasmuch as without the parts the whole is nothing. Vice versa, whatever rebounds to the benefit of the citizen extends also to the city, the nature of which is to extend benefits to the citizen. For example, whatever is beneficial to a dancer must, in so far as he is a dancer, be so also to the whole choric ballet. Applying this reasoning to the discursive power of the soul, it will shed light on every particular duty, and we shall never omit to perform whatever may by us be due to our country.

That is the reason why a man who proposes to act honorably by his country should from his soul remove every passion and disease. The laws of his country should, by a citizen, be observed as precepts of a secondary divinity, conforming himself entirely to their mandates. He who endeavors to transgress or make any innovation in these laws should be opposed in every way, and be prevented therefrom in every possible way. By no means beneficial to the city is contempt of existing laws and preference for the new. Incurable innovators, therefore, should be restrained from giving their votes, and making hasty innovations. I therefore commend the legislator Zaleucus of Locri, who ordained that he who intended to introduce a new law should do it with a rope around his neck, in order that he might be immediately strangled unless he succeeded in changing the ancient constitution of the state to the very great advantage of the community.

But customs which are truly those of the country and which, perhaps, are more ancient than the laws themselves, are, no less than the laws, to be preserved. However, the customs of the present, which are but of yesterday, and which have been everywhere introduced only so very recently, are not to be dignified as the institutes of our ancestors, and perhaps they are not even to be considered as customs at all. [1] Moreover, because custom is an unwritten law, it has as sanction the authority of a very good legislator, namely common consent of all that use it, and perhaps on this account its authority is next to that of justice itself.

3. On Proper Conduct Towards Our Parents

AFTER CONSIDERING THE GODS AND OUR COUNTRY, what person deserves to be mentioned more than, or prior to our parents? That is why we turn towards them. No mistake, therefore, will be made by him who says that they are as it were secondary or terrestrial divinities, since, on account of their proximity they should, in a certain non-blasphemous sense, be by us more honored than the Gods themselves. To begin with, the only gratitude worthy of the name is a perpetual and unremitting promptness to repay the benefits received from them, since, though we do our very utmost, this would yet fall short of what they deserve. Moreover, we might also say that in one sense our deeds are to be counted as theirs, because we who perform them were once produced by them. If, for instance, the works of Phidias and other artists should themselves produce other works of art, we should not hesitate to attribute these latter deeds also to the original artists; that is why we may justly say that our performances are the deeds of our parents, through whom we originally derived our existence.

In order that we may the more easily apprehend the duties we owe them, we should keep in mind the underlying principle that our parents should by us be considered as the images of the Gods, and, by Zeus, images of the domestic Gods, who are our benefactors, our relatives, our creditors, our lords, and our most stable friends. They are indeed most stable images of the Gods, possessing a likeness to them which no artist could possibly surpass. They are the guardian divinities of the home, and live with us; they are our greatest benefactors, endowing us with benefits of the greatest consequence, and indeed bestowing on us not only all we possess, but also such things as they wish to give us, and for which they themselves pray. Further they are our nearest kindred, and the causes of our alliance with others. They are also creditors of things of the most honorable nature and repay themselves only by taking what we shall be benefited by returning. For to a child what benefit can be so great as piety and gratitude to his parents? Most justly, too, are they our lords, for of what can we be the possession of in a greater degree than of those through whom we exist? Moreover, they are perpetual and spontaneous friends and auxiliaries, affording us assistance at all times and in every circumstance. Since, besides, the name of parent is the most excellent of names which we apply even to the divinities, we may add something to this conception: namely, that children should be persuaded that they dwell in their father's house, as if they were ministers and priests in a temple, appointed and consecrated for this purpose by nature herself, who entrusted to their care a reverential attention to their parents. If we are willing to carry out the dictates of reason we shall readily attend to both kinds of affective regard, for both the body and the soul. Yet reason will show us that to the body is to be paid less regard than to the soul, although we shall not neglect the former very necessary duties. For our parents, therefore, we should obtain liberal food, and such as is adequate to the weakness of old age; besides this, a bed, sleep, massage, a bath, and proper garments, in short, the necessities of the body, that they may at no time experience the want of any of these, by this imitating their care for the nurture of ourselves when we were infants. Our attention to them should partake of the prophetic nature, whereby we may discover what special bodily necessity that they may be longing for without expressing it to us. Respecting us, indeed, they divined many things when our desires could be expressed by no more than inarticulate and distressful cries, unable to express the objects of our wants clearly. By the benefits they formerly conferred upon us, our parents became to us the preceptors of what we ought to bestow upon them.

With respect to our parents' souls, we should in the first place procure for them diversion, which will be obtained especially if we associate with them by night and day, taking walks, being massaged, and living by their side, unless something necessary interferes. For just as those who are undertaking a long journey desire the presence of their families and friends to see them off, as if accompanying a solemn procession, so also parents, verging on the grave, enjoy most of all the diligent and unremitting attention of their children. Moreover, should our parents at any time, as happens often, especially with those whose education was deficient, display conduct which is reprehensible, they should indeed be corrected, but not as we are accustomed to do with our inferiors or equals, but as it were with suggestiveness -- not as if they had erred through ignorance, but as if they had committed an oversight through inattention, as if they would not have erred had they considered the matter. For reproof, especially if personal, is to the old very bitter. That is why their oversights should be supplemented by mild exhortation, as by an elegant artifice.

Children, besides, cause their parents to rejoice by performing for them servile offices such as washing their feet, making their bed, or ministering to their wants. These necessary servile attentions are all the more precious when performed by the dear hands of their children, accepting their ministrations. Parents will be especially gratified when their children publicly show their honor to those whom they love and very much esteem.

That is why children should affectionately love their parents' kindred and pay them proper attention, as also to their parents' friends and acquaintances. These general principles will aid us to deduce many other smaller filial duties which are neither unimportant nor accidental. For since our parents are gratified by the attention we pay to those they love, it will be evident that as we are in a most eminent degree beloved by our parents, we shall surely much please them by paying a proper attention to ourselves.

4. On Fraternal Love

THE FIRST ADMONITION, therefore, is very clear and convincing, and generally obligatory, being sane and self-evident. Here it is: Act by everyone in the same manner as if you supposed yourself to be him, and him to be you. A servant will be well treated by one who considers how he would like to be treated by him if he was the master, and himself the servant. The same principle might be applied between parents and children, and vice versa, and, in short, between all men. This principle, however, is peculiarly adapted to the mutual relation of brothers, since no other preliminary considerations are necessary, in the matter of conduct towards one's brother, than promptly to assume that equitable mutual relation. This therefore is the first precept, to act toward one's brother in the same manner in which he would think it proper for his brother to act towards him.

But someone will say, by Zeus, I do not transgress propriety and am equitable, but my brother's manners are rough and brusque. This is not right for, in the first place, he may not be speaking the truth, as excessive vanity might lead a man to extol and magnify his own manners and diminish and vilify what pertains to others. It frequently happens, indeed, that men of inferior worth prefer themselves to others who are far more excellent characters. Second, though the brother should indeed be of the rough character mentioned above, the course to take would be to prove oneself the better character by vanquishing his rusticity by your beneficence. Those who conduct themselves worthily towards moderate, gracious men are entitled to no great thanks; but to transform to graciousness the stupid vulgar man, he deserves the greatest applause.

It must not be thought impossible for exhortation to take marked effect, for in men of the most impossible manners there are possibilities of improvement, and of love and honor for their benefactors. Not even animals, and such as naturally are the most hostile to our race, who are captured by violence and dragged off in chains, and confined in cages, are beyond being tamed by appropriate treatment and daily food. Will not then the man who is a brother, or even the first man you meet, who deserves attention far greater than a beast, be rendered gentle by proper treatment, even though he should never entirely lose his boorishness? In our behavior, therefore, towards every man, and in a much greater degree towards our brother, we should imitate Socrates who, to a person who cried out against him, "May I die, unless I am revenged on you," answered, "May I die if I do not make you my friend!" So much then for external, fraternal relations.

Further, a man should consider that in a certain sense his brothers are part of him, just as my eyes are part of me; also my legs, my hands, and other parts of me. So are the relations of brother to a family social organism. If then the eyes and the hands should receive a particular soul and intellect, they would because of the above mentioned communion, and because they could not perform their proper offices without the presence of the other members who watch over the interests of the other members with the interest of a guardian spirit. So also, we who are men and who acknowledge that we have a soul should, towards our brothers, omit no proper offices. Indeed, more naturally adapted for mutual assistance than parts of the body are brothers. The eyes, being mutually adjusted, do see what is before them, and one hand cooperates with the other, but the mutual adaptation of brothers is far more various. For they accomplish things which are mutually profitable, though at the greatest intervening distance; and they will greatly benefit each other though their mutual differences be immeasurable. In short, it must be recognized that our life resembles nothing so much as a prolonged conflict which arises partly from the natural strife in the nature of things, and partly through the sudden unexpected blows of fortune, but most of all through vice itself, which abstains neither from violence, fraud, nor evil strategems. Hence nature, as not being ignorant of the purpose for which she generated us, produced each of us, as it were, accompanied by an auxiliary.

No one, therefore, is alone, nor does he derive his origin from an oak or a rock, but from parents, in conjunction with brothers, relatives, and other intimates. Here reason for us performs a great work, conciliating us to strangers who are no relatives of ours, furnishing us with many assistants. That is the very reason why we naturally endeavor to allure and make everyone our friend. How insane a thing it therefore is to wish to be united to those who naturally have nothing suitable to procure our love, and become as familiar as possible with them voluntarily, and yet neglect those willing helpers and associates supplied by nature herself, who are called brothers!

5. On Marriage

THE DISCUSSION OF MARRIAGE is most necessary as the whole of our race is naturally social, and the most fundamental social association is that effected by marriage. Without a household there could be no cities; and households of the unmarried are most imperfect, while on the contrary those of the married are most complete. That is why, in our treatise On Families, we have shown that the married state is to be preferred by the sage, while a single life is not to be chosen except under peculiar circumstances. Therefore, inasmuch as we should imitate the man of intellect so far as possible, and as for him marriage is preferable, it is evident it will be so also for us, except if hindered by some exceptional circumstance. This is the first reason for marriage.

Entirely apart from the model of the sage, Nature herself seems to incite us thereto. Not only did she make us gregarious, but adapted to sexual intercourse, and proposed the procreation of children and stability of life as the one and universal work of wedlock. Now Nature justly teaches us that a choice of such things as are fit should be made so as to accord with what she has procured for us. Every animal, therefore, lives in conformity to its natural constitution, and so also every plant lives in harmony with its laws of life. But there exists this difference, that the latter do not employ any reasoning or calculation in the selection of the things on which they lay hold, using nature along without participation in [rational] soul. Animals are drawn to investigate what may be proper for them by imagination and desires. To us, however, Nature gave reason to survey everything else and, together with all things -- nay, prior to all things -- to direct its attention to Nature itself, so as to tend towards her as a glorious aim, in an orderly manner, that by choosing everything consonant with her, we might live in a becoming manner. Following this line of argument, he will not error in saying that a family without wedlock is imperfect, for nature does not conceive of the governor without the governed, nor of the governed without a governor. Nature therefore seems to me to shame those who are adverse to marriage.

In the next place, marriage is beneficial. First, because it produces a truly divine fruit, the procreation of children who are, as partaking of our nature, to assist us in all our undertakings while our strength is yet undiminished; and when we shall be worn out, oppressed with old age, they will be our assistants. In prosperity they will be the associates of our joy and, in adversity, the sympathetic diminishers of our sorrows.

Marriage is beneficial not only because of procreation of children, but for the association of a wife. When we are wearied with our labors outside of the home, she receives us with officious kindness and refreshes us by her solicitous attentions. Next, she induces a forgetfulness of molestations outside of the house. The annoyances in the forum, the gymnasium, or the country, and in short all the vicissitudes of our intercourse with friends and acquaintances, do not disturb us so obviously, being obscured by our necessary occupations; but when released from these, we return home, and our mind has time to reflect, then availing themselves of this opportunity these cares and anxieties rush in upon us, to torment us, at the very moment when life seems cheerless and lonely. Then comes the wife as a great solace and, by making some inquiry about external affairs, or by referring to and together considering some domestic problem, she, by her sincere vivacity inspires him with pleasure and delight. It is needless to enumerate all the help a wife can be in festivals, when making sacrifice; or, during her husband's journeys, she can keep the household running smoothly, and direct at times of urgency; or in managing the domestics and in nursing her husband when sick.

In summary: in order to pass through life properly, all men need two things -- the aid of relatives, and kindly sympathy. But nothing can be more sympathetic than a wife, nor anything more kindred than children. Both of these are afforded by marriage; how therefore could be found anything more beneficial?

Also beautiful is a married life, it seems to me. What relation can be more ornamental to a family that that between husband and wife? Not sumptuous edifices, nor walls covered with marble plaster, nor piazzas adorned with stones, which indeed are admired by those ignorant of true goods; nor paintings and arched myrtle walks, nor anything else which is the subject of astonishment to the stupid, is the ornament of a family. The beauty of a household consists in the conjunction of man and wife, united to each other by destiny, and consecrated to the Gods presiding over nuptial birth and houses, and who harmonize, and use all things in common for their bodies, or even their very souls; who likewise exercise a becoming authority over their house and servants; who are properly solicitous about the education of their children; and to the necessities of life pay an attention which is neither excessive nor negligent, but moderate and appropriate. For, as the the most admirable Homer says, what can be more excellent

Than when at home the husband and wife
Live in entire unanimity.
(Odyssey, 7. 183).

That is the reason why I have frequently wondered at those who conceive that life in common with a woman must be burdensome and grievous. Though to them she appears to be a burden and molestation, she is not so; on the contrary, she is something light and easy to be borne or, rather, she possesses the power of charming away from her husband things burdensome and grievous. No trouble so great is there which cannot easily be borne by a husband and wife who harmonize and are willing to endure it in common. But what is truly burdensome and unbearable is impudence, for through it things naturally light, and among others a wife, become heavy.

To many, indeed, marriage is intolerable, in reality not from itself, or because such an association as this with a woman is naturally insufferable, but when we marry the wrong person and, in addition to this, are ourselves entirely ignorant of life, and unprepared to take a wife in such a way as a free-born woman ought to be taken, then indeed it happens that this association with her becomes difficult and intolerable. Vulgar people do marry in this way, taking a wife neither for the procreation of children, nor for harmonious association, being attracted to the union by the magnitude of the dowry, or through physical attractiveness, or the like; and by following these bad counsellors, they pay no attention to the bride's disposition and manners, celebrating nuptials to their own destruction, and with crowned doors introduce to themselves instead of a wife a tyrant, whom they cannot resist, and with whom they are unable to contend for chief authority.

Evidently, therefore, marriage becomes burdensome and intolerable to many, not through itself, but through these causes. But it is not wise to blame things which are not harmful, nor to make our own deficient use of these things the cause of our complaint against them. Most absurd, besides, is it feverishly to seek the auxiliaries of friendship, and achieve certain friends and associates to aid and defend us in the vicissitudes of life, without seeking and endeavoring to obtain the relief, defence and assistance afforded us by Nature, by the Gods, and by the laws, through a wife and children.

As to a numerous offspring, it is generally suitable to nature and marriage that all, or the majority of the offspring be nurtured. Many dissent from this, for a not very beautiful reason, through love of riches, and the fear of poverty as the greatest evil. To begin with, in procreating children we are not only begetting assistants, nurses for our old age, and associates in every vicissitude of life -- we do not however beget them for ourselves alone, but in many ways also for our parents. To them our procreation of children is gratifying because, if we should suffer anything calamitous prior to their decease we shall, instead of ourselves, leave our children as the support of their old age. Then for a grandfather is it a beautiful thing to be conducted by the hands of his grandchildren, and by them to be considered worthy of every attention. Hence, in the first place, we shall gratify our own parents by paying attention to the procreation of children. In the next, we shall be cooperating with the ardent wishes and fervent prayers of those who begot us. They were solicitous about our birth from the first, thereby looking for an extended succession of themselves, that they should leave behind them children of children, therefore paying attention to our marriage, procreation and nurture. Hence, by marrying and begetting children we shall be, as it were, fulfilling a part of their prayers; while, acting contrarily, we shall be destroying the object of their deliberate choice.

Moreover, it would seem that everyone who voluntarily, and without some prohibiting circumstance avoids marriage and the procreation of children, accuses his parents of madness, as having engaged in wedlock without the right conception of things. Here we see an unavoidable contradiction. How could that man live without dissension who finds a pleasure in living and willingly continues in life, as one who was properly brought into existence by his parents, and yet conceives that for him procreation of offspring is something to be rejected?

We must remember that we beget children not only for our own sake but, as we have already stated, for our parents'; but further also for the sake of our friends and kindred. It is gratifying to see children which are our offspring on account of human kindness, relatives, and security. Like ships which, though greatly agitated by the waves, are secured by many anchors, so do those who have children, or whose friends or relatives have them, ride at anchor in port in absolute security. For this reason, then, will a man who is a lover of his kindred and associates earnestly desire to marry and beget children.

Our country also loudly calls upon us to do so. For after all we do not beget children so much for ourselves as for our country, procuring a race that may follow us, and supplying the community with successors to ourselves. Hence the priest should realize that to the city he owes priests; the ruler, that he owes rulers; the orator, that he owes orators; and in short, the citizen, that he owes citizens. So it is gratifying to those who compose a choric ballet that it should continue perennially; and as an army looks to the continuance of its soldiers, so the perpetuation of its citizens is a matter of concern to a city. A city would not need succession were it only a temporary grouping, of duration commensurate with the lifetime of anyone man; but as it extends to many generations, and if it invokes a fortunate genius may endure for many ages, it is evidently necessary to direct its attention not only to its present, but also to its future, not despising our native soil: nor leaving it desolate, but establishing it in good hopes from our prosperity.

6. On Conduct Towards Our Relatives

DUTIES TO RELATIVES depend on duties to our immediate families, the arguments of which apply also to the former. Each of us is, indeed, as it were circumscribed by many circles, larger and smaller, comprehending and comprehended, according to various mutual circumstances.

The first and nearest circle is that which everyone describes about the center of his own mind, wherein is comprehended the body and all its interests; this is the smallest circle, nearly touching the center itself. The second and further circle which comprehends the first is that which includes parents, brethren, wife and children. The third greater circle is the one containing uncles, aunts, grandfathers and grandmothers, and the children of brothers and sisters. Beyond this is the circle containing the remaining relatives. Next to this is the circle containing the common people, then that which comprehends our tribe, then that of all the citizens; then follow two further circles: that of the neighboring suburbs, and those of the province. The outermost and greatest circle is that which comprehends the whole human race.

In view of this, he who strives to conduct himself properly in each of these connections should, in a certain respect, gather together the circles into one center, and always endeavor to transfer himself from the comprehending circles to the several particulars to which they comprehend.

The lover of his kindred, therefore, should conduct himself in a becoming manner towards his parents and brother; also, according to the same analogy, towards the more elderly of his relatives of both sexes, such as grandfathers, uncles and aunts; towards those of the same age as himself, as his cousins; and towards his juniors, as the children of his cousins. This summarizes his conduct towards his kindred, having already shown how he should act towards himself, toward his parents and brothers, and besides these, toward wife and children. To which must be added that those who belong to the third circle should be honored similarly to these, and again, kindred similarly to those that belong to the third circle. For benevolence must somehow fade away from those who are more distant from us by blood, though at the same time we should endeavor to effect a mutual assimilation. This distance will moderate if through the diligent attention which we pay to them we shorten the bond connecting us with each. Such then are the most comprehensive duties towards our kindred.

It might be well to say a word about the general names of kindred, such as the calling of cousins, uncles and aunts by the names of brothers, fathers and mothers; while of the other kindred, to call some uncles, others the children of brothers and sisters, and others cousins, according to the difference in age, for the sake of the emotional extension derivable from names. This mode of appellation will manifest our sedulous attention to these relatives, and at the same time, will incite and extend us in a greater degree to the contraction of the above circles.

We should however remember the distinction between parents that we made above. Comparing parents, we said that to mother was due more Jove, but to the father more honor. Similarly, we should show more love to those connected with us by a maternal alliance, but more honor to those connected with us by an alliance that is paternal.

7. On Economics [2]

TO BEGIN WITH, we must mention the kind of labor which preserves the union of the father. To the husband are usually assigned rural, forensic and political activities, while to the mother belong spinning of wool, making of bread, cooking, and in short, everything of a domestic nature. Nevertheless, neither should be entirely exempt from the labors of the other. For sometimes it will be proper, when the wife is in the country, that she should superintend the laborers and act as master of the household; and that the husband should sometimes attend to domestic affairs, inquiring about and inspecting what is doing in the house. This joint participation of necessary cares will more firmly unite their mutual association.

We should not fail to mention the manual operations which are associated with the spheres of occupations. Why should the man meddle with agricultural labors? This is generally admitted, and though men of the present day spend much time in idleness and luxury, yet it is rare to find any unwilling to engage in the labor of sowing and planting, and other agricultural pursuits. Much less persuasive perhaps will be the arguments which invite the man to engage in those other occupations that belong to the woman. For such men as pay little attention to neatness and cleanliness will not conceive wool-spinning to be their business since, for the most part, vile diminutive men, delicate and effeminate, apply themselves to the elaboration of wool, through an emulation of feminine softness. But it does not become a man who is manly to apply himself to things of this kind, so perhaps neither shall I advise such employments to those who have not unmistakably demonstrated their modesty and virility. What therefore should hinder the man from sharing in the labors pertaining to a woman, whose past life has been such as to free him from all suspicion of absurd and effeminate conduct? For is it not thought that more domestic labors pertain to man than to women in other fields? For they are more laborious, and require corporeal strength such as to grind, to knead meal, to cut wood, to draw water from a well, to carry large vessels from one place to another, to shake coverlets and carpets and the like. It will be quite proper for men to engage in such occupations.

But it would be well if the legitimate work of a woman be enlarged in other directions so that she may not only engage with her maid- servants in the spinning of wool, but may also apply herself to other more virile occupations. It seems to me that bread-making, drawing water from a well, the lighting of fires, the making of beds, and such like are labors suited to a free-born woman.

But to her husband a wife will seem much more beautiful, especially if she is young, and not yet worn out by the bearing of children, if she becomes his associate in the gathering of grapes and collecting the olives; and if he is verging toward old age, she will render herself more pleasing to him by sharing with him the labor of sowing and plowing, and while he is digging or planting, extending to him the instruments he needs for his work. For when by the husband and wife a family is governed thus, in respect to necessary labors, it seems to me that it will be conducted in the best possible manner.



1. Hierocles is referring to the innovations of the Christians.

2. The word "economy" is derived from oikos (house) and nomos (law).
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:02 am



TIMAEUS OF LOCRI is the central character of Plato's dialogue dealing with Pythagorean cosmology but it is uncertain whether or not there ever actually was a Timaeus of Locri who was a Pythagorean.

This particular writing may have gone through several stages of composition. It started out as an epitome of the cosmology of the Platonic dialogue. However, certain important details are not adequately treated and, in general, the cosmology is reduced to a series of statements while the underlying explanations and reasonings are omitted. At this stage the writing could well have been a summary of a student, and it seems likely that the attribution of the writing to Timaeus was added at a later date. Also added at a later date was a table of tone numbers which relate to the division of the world soul.

The best translation of this work is that of Thomas Tobin, published by Scholars Press in 1985, which features a good introduction and extensive notes. Tobin argues persuasively that this work may be seen as a Middle Platonic interpretation of the Timaeus; hence it was probably composed in the first century of, or before, the common era.

While Plato's Timaeus constitutes essential reading for anyone interested in Pythagorean cosmology, the writing reproduced here possesses more significance for the history of Middle Platonism than it does for the study of Pythagorean thought.


1. Mind, Necessity, Form and Matter

TIMAEUS OF LOCRI said the following:

Of all the things in the universe there are two causes: Mind, of things existing according to reason; and Necessity, of things [existing] by force, according to the power of bodies. The former of these causes is the nature of the good, and is called God, and the principle of things that are best, but what accessory causes follow are referred to Necessity. Regarding the things in the universe, there exist Form, Matter and the Perceptible which is, as it were, an offspring of the two others. Form is unproduced, unmoved, stationary, of the nature of the Same, perceptible by the mind, and a pattern of such things produced as exist by a state of change: that is what Form is said to be.

Matter, however. is a recipient of impressions, is a mother and a nurse, and is procreative of the third kind of being; for receiving upon itself the resemblances of form, and as it were remoulding them, it perfects these productions. He asserted moreover that matter, though eternal, is not unmoved; and though of itself it is formless and shapeless, yet it receives every kind of form; and that which is around bodies is divisible and partakes of the nature of the Different; and that matter is called by the twin names of Place and Space.

These two principles then are opposite to each other, of which Form is analogous to a male power and a father, while matter is analogous to a female power and a mother. The third thing is their offspring. Being three, they are recognizable by three marks: Form, by mind, according to knowledge; Matter by a spurious kind of reasoning, because it cannot be mentally perceived directly, but by analogy; and their production by sensation and opinion.

2. Creation of the World

BEFORE THE HEAVENS, then, there existed through reason Form and Matter, and the God who develops the best. But since the older surpasses the younger, and the ordered surpasses the orderless, the deity, being good -- on seeing that Matter receives Form, and is altered in every way, but without order -- found the necessity of organizing it, altering the undefined to the defined, so that the differences between bodies might be proportionately related, not receiving various alterations at random. He therefore made this world out of the whole of Matter, laying it down as a limit to the nature of being, through its containing in itself the rest of things, being one, only-begotten, perfect, endowed with soul and reason -- for these qualities are superior to the soulless and the irrational -- and of a sphere-like body, for this is more perfect than the rest of forms.

Desirous then of making a very good production he made it a divinity, created and never to be destroyed by any cause other than the God who had put it in order, if indeed he should ever wish to dissolve it. But on the part of the good there is no rushing forward to the destruction of a very beautiful production. Such therefore being the world, it continues without corruption and destruction, being blessed. It is the best of things created, since it has been produced by the best cause, which looked not to patterns made by hand but to Form in the abstract, and to Existence, perceiving by the mind to which the created thing, having been carefully adjusted, has become the most beautiful. It is even perfect in the realm of sense because its pattern, containing in itself all the living things perceived by mind, left out nothing, being the limit of the things perceived by mind, as this world is of those perceived by sense.

Being solid and perceptible by touch and sight, the world has a share of earth and fire, and of the things between them, air and water; and it is composed of all perfect bodies, which are in it as wholes, so that no part might ever be left out, in order that the body of the Universe might be altogether self-sufficient, uninjured by corruption from without or within; for apart from these there is nothing else, and hence the things combined according to the best proportions and with equal powers, neither rule over, nor are ruled by each other in turn, so that some receive an increase, others a decrease, remaining indissolubly united according to the very best proportions.

3. Proportions of the World Combination

WHENEVER THERE ARE any three terms with mutually equal intervals that are proportionate, we then perceive that, after the matter of an extended string, the middle is to the first, as is the third to it, and this holds true inversely and alternately, interchanging places and order, so that it is impossible to arrange them numerically without producing an equivalence of results. Likewise the world's shape and movement are well arranged; the shape is a sphere, self-similar on all sides, able to contain all shapes that are similar, and the movement endlessly exhibits the change dependent on a circle. Now as the sphere is on every side equidistant from the center, it is able to retain its poise whether in movement or at rest, neither losing its poise nor assuming another. Its external appearance being exactly smooth, it needs no mortal organs such as are fitted to and present in all other living beings because of their wants. The world soul's element of divinity radiates out from the center entirely penetrating the whole world, forming a single mixture of divided substance with undivided form; and this mixture of two forces, the Same and the Different, became the origin of motion, which indeed was not accomplished in the easiest way, being extremely difficult.

Now all these proportions are combined harmonically according to numbers, which proportions were scientifically divided according to a scale which reveals the elements and the means of the soul's combination. Now seeing that the earlier is more powerful in power and time than the later, the deity did not rank the soul after the substance of the body, but made it older by taking the first of unities, 384. Knowing this first, we can easily reckon the double (square) and the triple (cube); and all the terms together, with the complements and eights, there must be 36 divisions, and the total amounts to 114,695.

These are the divisions

FIGURE 15. A TABLE OF TONE NUMBERS. There is evidence that this table of tone numbers is a later addition to the text. The abbreviations I. and ap. represent the two types of semitones, the leimma and the apotome respectively.

4. Planetary Revolutions and Time

GOD THE ETERNAL, the chief ruler of the Universe and its creator, is beheld alone by the mind, but we may behold by sight all that is produced in this world in connection with those parts which are heavenly, and, being ethereal, must be divided into kinds: some relating to Sameness, others to Difference. Sameness draws onward all that is within, with the general motion of the entire sphere of the universe from east to west. Difference draws along all self-moved portions from west to east, fortuitously rolled around and along by the superior power of Sameness. [1]

The movement of the Different, being divided in harmonic proportion, assumes the order of seven circles. Nearest to the earth the Moon revolves in a month, while beyond her the Sun completes his revolution in a year. Two planets run co-equal with that of the sun: Mercury and the star of Hera, the latter of which is also called the star of Venus and the Lightbringer because shepherds and common people, generally not skillful in sacred astronomy, confuse the western and eastern risings. The same star may shine in the west when following the sun at a distance great enough to be viewed in spite of solar splendor; and at another time in the east when, as herald of the day, it rises before the sun, leading it. Because of its running together with the sun, Venus is the Lightbringer frequently but not always; for there are planets and stars of any magnitude seen above the horizon before sunrise, herald of the day. But the three other planets, Mars, Jupiter and Saturn have their peculiar velocities and different years, completing their course while making their periods of effulgence, of visibility, of obscuration and eclipse, accurately rising and setting. Moreover, they complete their appearances conspicuously in the east or west according to their position relative to the sun who, during the day, speeds westward, which during the night it reverses, under the influence of Sameness, while its annual revolution is due to its inherent motion. In consequence of these two kinds of motion it roils out a spiral, moving one degree each day, but is whirled around under the sphere of the fixed stars according to each revolution of darkness and day.

Now these revolutions are by men called portions of time, which the deity arranged together with the world. For before the world the stars did not exist, and hence there was neither year, nor periods of season, by which this generated time is measured, and which is the representation of the ungenerated time called eternity. For as this heaven has been produced according to an eternal pattern, the world of ideas, so was our world-time created simultaneously according to the pattern of eternity.

5. The World's Creation by Geometric Figures

THE EARTH, fixed at the center, becomes the hearth of the Gods and the boundary of darkness and day, producing settings and risings according to the occultation produced by the things that form the boundary, just as we improve our sight by making a tube with our closed hand, to exclude refraction. Earth is the oldest body in the heavens. Water was not produced without earth, nor air without moisture, nor could fire continue without moisture and the materials that are inflammable, so that earth is fixed upon its balance at the root and base of all other substances.

Of produced things, the substratum is Matter, while the reason of each shape is abstract Form; of these two the offspring is Earth and Water, Air and Fire.

This is how they were created. Every body is composed of surfaces, whose elements are triangles, of which one is right-angled, and the other has all unequal sides, with the square of the longer side being thrice the size of the lesser, while its least angle is the third of a right angle. The middle one is the double of the least, for it is two-thirds of a right angle. The greatest is a right angle, being one-and-a-half times greater than the middle one, and the triple of the least. Now this unequal sided triangle is half of an equilateral triangle, cut into two equal parts by a line let down from the apex to the base. In each of these triangles there is a right angle; but in the one, the two sides about the right angle are equal, and in the other all the sides are unequal. Now let this be called a scalene triangle; while the other, the half of the square, is the principle of the constitution of Earth. For the square produced from this scalene triangle is composed of four half-squares and from such a square is produced the cube, the most stationary and steady form in every way, having six sides and eight angles. On this account Earth is the heaviest and most difficult elemental body to move, and its substance is inconvertible, because it has no affinity with the other type of triangle. Only Earth has a peculiar element of the square, while the other triangle is the element of the three other substances, Fire, Air and Water. For when the half triangle is put together six times it produces the solid equilateral triangle, the exemplar of the tetrahedron, which has four faces with equal angles, which is the form of Fire, as the easiest to be moved, and composed of the finest particles. After this ranks the octahedron, with eight faces and six angles, being the element of Air; and the third is the icosahedron, with twenty faces and twelve angles, being the element of Water, composed of the most numerous and heaviest particles.

These then, as being composed of the same element, are transmuted into one another. But the deity made the dodecahedron the image of the Universe, as being the nearest to the sphere. Fire then, by the fineness of its particles, passes through all things; and Air through the rest of things, with the exception of Fire; and Water through the Earth. All things are therefore full, and have no vacuum. They cohere by the revolving movement of the Universe, and are pressed against and rubbed by each other in turn, and produce the never- failing change from generation to destruction.

6. Concretion of the Elements

BY MAKING USE OF THESE THE DEITY put together this world, sensible to touch through the particles of Earth, and to sight through those of Fire, which two are the extremes. Through the particles of Air and Water he had conjoined the world by the strongest chain, namely proportion, which restrains not only itself but all its subjects. Now if the conjoined object is a plane surface one middle term is sufficient, but if a solid there will be need of two mean terms. With two middle terms, therefore, he combined two extremes, so that as Fire is to Air, Air is to Water; and as Air is to Water, so is Water to Earth; and by alternation, as Fire is to Water, Air might be to Earth. Now since all are equal in power, their ratios are in a state of equilibrium. This world then is one, through the bond of the deity, made according to proportion.

Now each of these substances possesses many forms. Fire has those of flame, burning and luminousness, through the inequality of the triangles in each of them. In the same manner Air is partly clear and dry, and partly turbid and foggy. Water can be partly flowing and partly congealed, as it is in snow, frost, hail or ice. That which is moist is in one respect flowing as honey and oil, but in another is compact as pitch and wax. Concerning solids some are fusible, as gold, silver, copper, tin, lead and copper, and some brittle as sulphur, asphalt, nitre, salt, alum, and similar materials.

7. Composition of the Soul

AFTER PUTTING TOGETHER THE WORLD, the deity planned the creation of mortal beings so that, himself being perfect, he might perfectly complete the world. Therefore he mixed up the soul of man out of the same proportions and powers, and after taking the particles and distributing them, he delivered them over to Nature, whose office is to effect change. She then took up the task of working out mortal and ephemeral living beings, whose souls were drawn in from different sources, some from the moon, others from the sun, and others from various planets, [from] that cycle within the Difference, with the exception of one single power which was derived from Sameness, which she mixed up in the rational portion of the soul, as the image of wisdom in those of a happy fate.

Now in the soul of man one portion is rational and intellectual, and another irrational and unintellectual. Of the logical part the best portion is derived from Sameness, while the worst comes from Difference, and each is situated around the head so that the other parts of the soul and body may minister to it, as the supreme part of the whole body. Of the irrational portion, that which represents passion hovers around the heart, while desire inhabits the liver. The principle of the body and root of the marrow is the brain, wherein resides the ruling power; and from this, like an effusion, through the back-bone flows what is left over from the brain, from which are separated the particles of semen and seed. The marrow's surrounding defenses are the bones, of which the flesh is the covering and concealment. To the nerves he united joints by tendons, suitable for their movement. Of the internal organs, some exist for the sake of nourishment and others for safety. Of exterior motions, some are conveyed to the interior intelligent places of perception while others, not falling under the power of apprehension, are unperceived, either because the affected bodies are too earth-like or because the movements are too feeble. The painful movements tend to arouse nature, while the pleasurable lull nature into remaining within herself.

8. Sensations

AMONG THE SENSES, the deity has in us lit sight to view the objects in the heavens and for the reception of knowledge, while to make us capable to receive speech and melody, he has implanted hearing in us, of which he who is deprived thereof from birth will become dumb, nor be able to utter any speech, and that is why this sense is said to be related closest to speech.

Many affections of the body that have a name are so called with reference to touch, and others from relation to their seat. Touch judges the properties connected with life such as warmth, coldness, dryness, moisture, smoothness, roughness, and of things that are yielding, opposing, hard or soft. Touch also judges heaviness or lightness. Reason defines these affections as being centripetal and centrifugal, which men mean to express when they say below and middle. For the center of a sphere is below, and that part lying above it and stretching to the circumference is called upwardness.

Now what is warm appears to consist of fine particles, causing bodies to separate, while coldness consists of the grossness of the particles, causing a tendency to condense.

The circumstances connected with the sense of taste are similar to those of touch. For substances grow either astringent or smooth through contraction and dilation, also by entering the pores and assuming shapes. For those that cause the tongue to melt away, or that scrape it, appear to be rough. Those that act moderately in scraping appear brackish. Those that inflame or separate the skin are acrid; while their opposites, the smooth and sweet, are reduced to a juicy state.

The kinds of odors have not been defined because they percolate through narrow pores that are too stiff to be closed or separated. Things seem to be sweet smelling or bad smelling from the putrefaction or concoction of the earth and of similar substances.

A vocal sound is a percussion in the air, arriving at the soul through the ears. The passages of the ears reach to the liver, and among them is pneuma, by the movement of which hearing exists. Now of the voice and hearing, that portion which is quick is acute, while that which is slow is grave, the medium being the most harmonious. What is much and diffused is great; what is little and compressed is small; what is arranged to musical proportions is in tune; while that which is unarranged and unproportionate is out of tune and not properly adjusted.

The fourth kind of things relating to the senses is the most uniform and various, and they are called objects of sight, in which are all kinds of colors and an infinity of colored substances. The principle ones are four: white. black, brilliant [blue] and red, out of a mixture of which all other colors are prepared. What is white dilates the eye, and what is black causes it to contract, just as warmth expands, and cold contracts, and what is rough contracts the tasting, and what is sharp dilates it.

9. Respiration

IT IS NATURAL for the covering of animals that live in the air to be nourished and kept together by the food being distributed in the veins through the whole mass, in the manner of a stream, conveyed as it were by channels, and moistened by the breath, which diffuses it, and carries it to the extremities. Respiration is produced because there is no vacuum in nature: the air, as it flows in, is inhaled in place of that which is exhaled through unseen pores, such as those through which perspiration drops appear on the skin, but a portion is excreted by the natural warmth of the body. It then becomes necessary for an equivalent portion to be reintroduced to avoid a vacuum.

Now in lifeless substances, according to the analogy of respiration, the same organization occurs. The cupping-glass and amber, for instance, bear resemblance to respiration. [2]

Now the breath flows through the body to an orifice outwards and is, in turn, introduced through respiration by the mouth and nostrils, and again after the manner of the flow of the Euripus, is in turn carried to the body which is extended according to the expiration. Also the cupping-glass, when the air within is expelled by fire, attracts moisture to itself; and amber, when the air is separated from it, attracts a nearby substance.

Now all nourishment comes as from a root from the heart, and from the stomach as a fountain, and is conveyed to the body to which, if it be moistened by more than what flows out, there is said to be an increase, but if more flows out it is known as a decay. The point of perfection is the boundary between these two, and is considered to exist in an equality of the efflux and influx; but when the joints of the system are broken, should there no longer exist any passage for the breath, or the nourishment should not be distributed, then the animal dies.

10. Disorders

THERE ARE MANY THINGS HURTFUL TO LIFE which are the causes of death. One kind is disease. Its beginning is disharmony of the functions when the simple powers such as heat, cold, moisture or dryness are excessive or deficient. Then come changes and alterations in the blood from corruption and the deterioration of the flesh, when changes in the blood or flesh take place according to the changes of what is acid, brackish, or pungent in the blood. Hence arise the production of bile and of phlegm, of diseased juices, and of rotten liquids. Their effects are weak indeed, unless deeply seated, but difficult to cure when their commencement is generated from the bones, and acute if located in the marrow. The last disorders are those of the breath, bile and phlegm, when they increase and flow into places inappropriate for them. For by taking the place of the better, and driving away what is congenial, they fix themselves there, injuring bodies, and resolving these into themselves.

These then are the sufferings of the body, and hence arise many diseases of the soul, some from one faculty, some from another. Of the perceptive soul the disease is a difficulty of perception; of the recollecting, a forgetfulness; of the appetitive part, a deficiency of desire and eagerness; of the affective, a violent suffering and excited madness; of the rational, an indisposition to learn and think.

But of wickedness the beginnings are pleasures and pains; desires and fears, inflamed by the body, mingled with the mind, and are called by different names. For there loves and desires let loose uncontrolled passions, heavy resentments, and appetites of various kinds, and immoderate pleasures. Plainly, to be unreasonably disposed towards the affections is the limit of virtue, and to be under their rule is that of vice; for to abound in them, or to be superior to them, places us in a good or bad position. Against such impulses the temperaments of our bodies are greatly able to cooperate whether quick or hot, or various, by leading us to melancholy or violent lewdness; and certain parts, when affected by a flux, produce itchings and forms of body more similar to a state of inflammation than one of health, through which a sinking of the spirits and a forgetfulness, delirium and a state of fear result.

11. Discipline

IMPORTANT, TOO, ARE THE HABITS in which persons are trained, in the city or at home, and their daily food, by luxury enervating the soul, or fortifying it for strength. For living out of doors, and simple fare, and gymnastic exercises, and the morals of companions, produce the greatest effect in the way of vice and virtue. These causes are derived from our parents and the elements, rather than ourselves, provided that on our part there be no remissness by keeping aloof from acts of duty. The animal cannot be in good condition unless the body possesses the better properties under its control, namely health and correct perception, and strength and beauty. Now the principles of beauty are a symmetry as regards its parts, and as regards the soul. For nature has arranged the body, like an instrument, to be subservient to and in harmony with the subjects of life. The soul must likewise be brought into harmony with its analogous good qualities, namely in the case of temperance, as the body is in the case of health; and in that of prudence, as in the case of correct perception; and in that of fortitude, as in the case of vigor and strength; and in that of justice, as in the case of beauty.

Nature, of course, furnishes their beginnings, but their continuation and maturation result from carefulness: those relating to the body from the gymnastic and medical arts, those to the soul through instruction and philosophy. For these are the powers that nourish and give a tone to the body and soul by means of labor and gymnastic exercise and pureness of diet; some through drug medication applied to the body, and others through discipline applied to the soul by means of punishments and reproaches, for by the encouragement they give strength and excite to an onward movement and exhort to beneficial deeds. The art of the gymnasium trainer and its nearest approximation, that of the medical man, do on application to the body reduce their powers to the utmost symmetry, purifying the blood and equalizing the breath so that, if there were any diseased virulence there, the powers of blood and breath may be made vigorous; but music and its leader philosophy which the Gods ordained as regulators for the soul, accustom, persuade, and partly compel the irrational to obey reason, and induce the spirited and appetitive parts of the soul to become one mild and the other quiet, so as not to be moved without reason, nor to be unmoved when the mind incites either to desire or enjoy something; for this is the definition of temperance, namely, docility and firmness. Intelligence and philosophy the highest in honor, after cleansing the soul from false opinions, have introduced knowledge, recalling the mind from excessive ignorance and setting it free for the contemplation of divine things, in which to occupy oneself with self-sufficiency, as regards the affairs of a man, and with an abundance, for the commensurate period of life, is a happy state.

12. Human Destiny

NOW HE WHOM THE deity has happened to assign somewhat of a good fate is, through opinion, led to the happiest life. But if he be morose and indocile, let the punishment that comes from law and reason follow him, bringing with it the fears ever on the increase, both those that originate in heaven or Hades, how that punishments inexorable are below laid up for the unhappy, as well as those ancient Homeric threats of retaliation for the wickedness of those defiled by crime. For as we sometimes restore bodies to health by means of diseased substances, if they will not yield to the more healthy, so if the soul will not be led by true reasoning, we restrain it by false. These are unusual since, by a change, we say that the souls of cowards enter into the bodies of women who are inclined to insulting conduct; and the souls of the blood-stained take on the bodies of wild beasts; the lascivious enter into the bodies of sows and boars; the light-minded and frivolous take on shapes of aeronautic birds; and those who neither learn nor think of anything enter into the bodies of idle fish.

On all these matters, however, there has at a second period been delivered a judgment by Nemesis, of Fate, together with the avenging deities that preside over murderers, and those under the earth in Hades, and the inspectors of human affairs to whom God, the leader of all, has entrusted the administration of the world, which is filled with Gods and men, and the rest of the living beings who have been fashioned according to the best model of an unbegotten, eternal and mentally-perceived form.



1. Plato identified the power of Sameness with the regular movement of the stars and the celestial sphere. The power of Difference is identified with the movements of the planets which, over the course of the year, wander in the opposite direction of the fixed stars.

2. As Thomas H. Tobin observes in his translation of this text, the cupping-glass was a device used in medicine. When heated the air inside was expelled and the device was placed on a wound to remove poison. Rubbing amber with a cloth was likewise thought to expell the air in it, causing the amber to attract small object to it as the vacuum was filled. The actual cause is static electricity.
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:03 am


IT WAS NOT UNCOMMON for the early church fathers to refer to the philosophers of Greece, even though for many the contact with their thought was only through outlines and handbooks rather than through primary sources.

For the most part the church fathers had positive things to say about the Pythagoreans. Clement of Alexandria, who had more than a passing acquaintance with Platonic thought, shows an interest in Pythagorean mathematics, and even applied the numbers of the harmonic ratio to the interpretation of scripture. Moreover, Justin Martyr sought entrance to a Pythagorean school but was rejected on account of inadequate mathematical knowledge; he later became a Platonist and then a Christian. The fragments which follow, which were not included in Guthrie's original edition of this book, are representative of what the church fathers had to say.


HERACLEIDES AND THE PYTHAGOREANS think: that each of the stars is a world, including an Earth, and an atmosphere and an ether in the infinite space. These doctrines are introduced in the Orphic Hymns, for they make each star a world. -- Ps.-Plutarch, On the Opinions of the Philosophers. Quoted by Eusebius, Preparation for the Gospel, 839b.

Then, in regular succession from another starting-point, Pythagoras the Samian, son of Mnesarchus, calls numbers, with their proportions and harmonies, and the elements composed of both, the first principles; and he includes also Unity and the Indefinite Dyad. - Justin Martyr, Exhortatory Address to the Greeks, IV.

For the Pythagorean Theano writes, "Life would indeed be a feast to the wicked, who, having done evil, then die; were not the soul immortal, death would be a godsend." -- Clement of Alexandria, Stromateis IV, 7.

Did not Theano the Pythagorean make such progress in philosophy, that to him who looked intently at her, and said, "Your arm is beautiful," she answered "Yes, but it is not public." Characterized by the same propriety there is also reported the following reply. When asked when a woman after being with her husband may attend a religious festival, said, "From her own husband at once, from a stranger never." -- Clement of Alexandria, Stromateis, IV, 19.

Pythagoras thus defined the being of God, "as a soul passing to and fro, and diffused through all parts of the universe, and through all nature, from which all living creatures which are produced derive their life." -- Lactantius, The Divine Institutes, I, 5.

And Pythagoras, son of Mnesarchus, who expounded the doctrines of his own philosophy mystically by means of symbols, as those who have written his life show, himself seems to have entertained thoughts about the unity of God not unworthy of his foreign residence in Egypt. For when he says that Unity is the first principle of all things, and that it is the cause of all good, he teaches by an allegory that God is one, and alone. And that this is so, is evident from his saying that unity and one differ widely from one another. For he says that unity belongs to the class of things perceived by the mind, but that one belongs to numbers. And if you desire to see a clearer proof of the opinion of Pythagoras concerning one God, hear his own opinion, for he spoke as follows: "God is one; and he himself does not, as some suppose, exist outside the world, but in it, he being wholly present in the entire circle, and beholding all generations, being the regulating ingredient of all the ages, and the administrator of his own powers and works, the first principle of all things, the light of heaven, and father of all, the intelligence and animating soul of the universe, the movement of all orbits." Thus, then, Pythagoras. -- Justin Martyr, Exhortatory Address to the Greeks, XIX.
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:04 am





Phaedo 62 B. The saying uttered in secret rites, to the effect that we men are in a sort of prison, and that one ought not to release himself from it nor yet to run away, seems to me something great and not easy to see through; but this at least I think is well said, that it is the Gods who care for us, and we men are one of the possessions of the Gods.

Crarylus 400 B. For some say that it [the body] is the tomb of the soul -- I think it was the followers of Orpheus in particular who introduced this word -- which has the soul enclosed like a prison in order that it may be kept safe.

Gorgias 493 A. I once heard one of the wise men say that now we are dead and the body (soma) is our tomb (sema), and that the part of the soul where desires are, it so happens, is open to persuasion, and moves upward or downward. And, indeed, a clever man -- perhaps some inhabitant in Sicily or Italy -- speaking allegorically, and taking the word from 'credible' (pithanos) and 'persuadable' (pistikos), called this a jar (pithos); and he called those without intelligence uninitiated, and that part of the soul of uninitiated persons where the desires are, he called its intemperatness, and said it was not water-tight, as a jar might be pierced with holes -- using the simile because of insatiate desires.

Gorgias 507 E. And the wise men say that one community embraces heaven and earth and Gods and men and friendship and order and temperance and righteousness, and for that reason they call this whole a kosmos, my friend, for it is not without order nor yet is there excess. It seems to me that you do not pay attention to these things, though you are wise in regard to them. But it has escaped your notice that geometrical equality prevails widely among both Gods and men.


Physics. iii. 4; 203 a I. Some, like the Pythagoreans and Plato, have made the Unlimited a first principle existing by itself, not connected with anything else, but being the infinite itself in its essence. Only the Pythagoreans found it among all things perceived by sense (for they say that number is not an abstraction), and they held that what is outside the heavens is Unlimited.

iii. 4; 203 a 10. The Pythagoreans identify the Unlimited with the even, For this, they say, is cut off and shut in by the odd, and provides things with an element of infinity. An indication of this is what happens with numbers. If gnomons are place round the one, and without the one, in the one construction the figure that results is always the same [square], in the other it is always different [oblong].

iii. 4; 204 a 33. [The Pythagoreans] both hold that the infinite is substance, and divide it into parts.

iv. 6; 213 b 22. And the Pythagoreans say that there is a void, and that it enters into the heaven itself from the infinite air, as though it [the heaven] were breathing; and this void defines the natures of things, inasmuch as it is a certain separation and definition of things that lie together; and this is true first in the case of numbers, for the void defines the nature of these.

On the Heavens. i. 1; 268 a 10. For as the Pythagoreans say, the All and all things are defined by threes; for end and middle and beginning constitute the number of the All, and also the number of the Triad.

ii. 2; 284 b 6. And since there are some who say that there is a right and a left of the heavens, as, for instance, those that are called Pythagoreans (for such is their doctrine), we must investigate whether it is as they say.

ii. 2; 285 a 10. Wherefore one of the Pythagoreans might be surprised in that they say that there are only these two first principles, the right and the left, and they pass over four of them as not having the least validity; for there is no less difference up and down, and front and back than there is right and left in all creatures.

ii. 2; 285 b 23. And some are dwelling in the upper hemisphere and to the right, while we dwell below and to the left, which is the opposite to what the Pythagoreans say; for they put us above and to the right, while the others are below and at the left.

ii. 9; 290 b 15. Some think it necessary that noise should arise when so great bodies are in motion, since sound does arise from bodies among us which are not so large and do not move so swiftly; and from the sun and moon and from the stars in so great number, and of so great size, moving so swiftly, there must necessarily arise a sound inconceivably great. Assuming these things and that the swiftness has the principle of harmony by reason of the intervals, they say that the sound of the stars moving on in a circle becomes musical. And since it seems unreasonable that we also do not hear this sound, they say that the reason for this is that the sound exists in the very nature of things, so as not to be distinguishable from the opposite silence; for the distinction of sound and silence lies in their contrast with each other, so that as blacksmiths think there is no difference between them because they are accustomed to the sound, so the same thing happens to men.

ii. 9; 291 a 7. What occasions the difficulty and makes the Pythagoreans say that there is a harmony of the bodies as they move, is a proof. For whatever things move themselves make a sound and noise; but whatever things are fastened in what moves or exist in it as the parts in a ship, cannot make a noise, nor yet does the ship if it moves in a river.

ii. 12; 293 a 19. They say that the whole heaven is limited, the opposite to what those of Italy, called the Pythagoreans, say; for these say that fire is at the center and that the earth is one of the stars, and that moving in a circle about the center it produces night and day. And they assume yet another earth opposite this which they call the counter-earth (antichthon), not seeking reasons and causes for phenomena, but stretching phenomena to meet certain assumptions and opinions of theirs and attempting to arrange them in a system....And what is more, the Pythagoreans say that the most authoritative part of the All stands guard, because it is specially fitting that it should, and this part is the center; and this place that the fire occupies, they call the Guardpost of Zeus, as it is called simply the center, that is, the center of space and the center of matter and of nature.

Metaphysics. i. 5; 985 b 23-986 b 8. With these before them [Anaxagoras, Empedocles, Atomists] those called Pythagoreans, applying themselves to the sciences, first developed them; and being brought up in them they thought that the first principles of these [i.e., numbers] were the first principles of all things. And since of these [sciences] numbers are by nature the first, in numbers rather than in fire and earth and water they thought they saw many likenesses to things that are and that are coming to be, as, for instance, justice is such a property of numbers, and soul and mind are such a property, and another is opportunity, and of other things one may say the same of each one.

And further, discerning in numbers the conditions and reasons of harmonies also -- since, moreover, other things seemed to be like numbers in their entire nature, and numbers were the first of every nature -- they assumed that the elements of numbers were the elements of all things, and that the whole heavens were harmony and number. And whatever characteristics in numbers and harmonies they could show were in agreement with the properties of the heavens and its parts and with its whole arrangement, these they collected and adapted; and if there chanced to be any gap anywhere, they eagerly sought that the whole system might be connected with these [stray phenomena]. To give an example of my meaning: inasmuch as ten seemed to be the perfect number and to embrace the whole nature of numbers, they asserted that the number of bodies moving through the heavens were ten, and when only nine were visible, for the reason just stated they postulated the counter-earth as the tenth. We have given a more definite account of these thinkers in other parts of our writings. But we have referred to them here with this purpose in view, that we might ascertain from them what they asserted as the first principles and in what manner they came upon the causes that have been enumerated. They certainly seem to consider Number as the first principle and, as it were, the matter in things and in their conditions and states; and the odd and the even are elements of number, and of these the one is Limited and the other Unlimited, and unity is the product of both of them, for it is both odd and even, and Number arises from the one, and the whole heaven, as has been said, is numbers.

A different party in this same school says that the first principles are ten, named according to the following table: Limited and Unlimited, Odd and Even, One and Many, Right and Left, Male and Female, Rest and Motion, Straight and Crooked, Light and Darkness, Good and Bad, Square and Oblong. After this manner Alcmaeon of Croton seems to have conceived them, and either he received this doctrine from them or they from him, for Alcmaeon arrived at maturity when Pythagoras was an old man, and his teachings resembled theirs. For he says that most human affairs are twofold, not meaning opposites reached by definition, as did the former party, but opposites by chance -- as, for example, white-black, sweet-bitter, good-bad, small-great. This philosopher let fall his indefinite opinions about the other contraries, but the Pythagoreans declared the number of the opposites and what they were. From both schools one may learn this much: that opposites are the first principles of things -- but from the latter he may learn the number of these, and what they are. Yet how it is possible to bring them into relation with the causes of which we have spoken they have not clearly worked out. They seem to range their elements under the category of matter, for they say that substance is compounded and formed from them, and that they inhere in it.

987 a 9-27. Down to the Italian philosophers, and with their exception, the rest have spoken more reasonably about these principles, except that, as we said, they do indeed use two principles, and the one of these, whence is motion, some regard as one and others as twofold. The Pythagoreans, however, while they in similar manner assume two first principles, add this which is peculiar to themselves: that they do not think that the Limited and the Unlimited and the One are certain other things by nature, such as fire or earth or any other such thing, but the Unlimited itself and Unity itself are the essence of things of which they are predicated, and so they make Number the essence of all things. So they taught after this manner about them, and began to discourse and to define what essence is, but they made it altogether too simple a matter. For they made their definition superficially, and to whatever first the definition might apply, this they thought to be the essence of the matter, as if one should say that twofold and two were the same, because the twofold subsists in the two. But undoubtedly the two and the twofold are not the same, otherwise one thing will be many -- a consequence which they actually drew. So much then may be learned from the earlier philosophers and from their successors.

i. 6; 987 b 10. And Plato only changed the name, for the Pythagoreans say that things exist by 'imitation' of numbers, but Plato, by 'participation.'

i. 6; 987 b 22. Plato concurred with the Pythagoreans in saying that the One is the real essence of things, and not something else with unity as an attribute. In harmony with them he affirms that Numbers are the principles of being for other things. But it is peculiar to him that instead of a single Indefinite he posits a double Indefinite, an Infinite of greatness and of littleness; and it is also peculiar to him that he separates Numbers from things that are seen, while they say that Numbers are the things themselves, and do not interpose mathematical objects between them. This separation of the One and Numbers from things, in contrast with the position of the Pythagoreans, and the introduction of Forms, are the consequence of his investigation by definitions.

i. 8; 989 b 32-990 a 32. Those, however, who carry on their investigation with reference to all things, and divide things into what are perceived and what are not perceived by sense, evidently examine both classes, so one must delay a little longer over what they say. They speak correctly and incorrectly in reference to the questions now before us. Now those who are called Pythagoreans use principles and elements yet stranger than those of the physicists, in that they do not take them from the sphere of sense, for mathematical objects are without motion, except in the case of astronomy. Still, they discourse about everything in nature and study it; they construct the heaven, they observe what happens in its parts and their states and motions; they apply to these their first principles and causes, as though they agreed entirely with the other physicists in that being is only what is perceptible and all that which is called heaven includes. But their causes and first principles, they say, are such as to lead up to the higher parts of reality, and are in harmony with this rather than with the doctrines of nature. In what manner motion will take place when Limit and Unlimited, Odd and Even, are the only underlying realities, they do not say; nor how it is possible for genesis and destruction to take place without motion and change, or for the heavenly bodies to revolve. Further, if one grants to them that spatial magnitude arises from these principles, or if this could be proved, still, how will it be that some bodies are light and some heavy? for their postulates and statements apply no more to mathematical objects than to things of sense; accordingly they have said nothing at all about fire or earth or any such objects, because I think they have no distinctive doctrine about things of sense. What is more, how is it necessary to assume that Number and states of Number are the causes of what is in the heavens and what is taking place there from the beginning and now, and that there is no other number than that out of which the world is composed? For when opinion and opportune time are at a certain point in the heavens, and a little farther up or down are injustice and judgment or a mixture of them, and they bring forward as proof that each one of these is Number, and the result then is that at this place there is already a multitude of compounded qualities because those states of Number have each their place -- is this Number in heaven the same which it is necessary to assume that each of these things is, or is it something different? Plato says it is different; still, he thinks that both these things and the cause of them are Numbers, but the one class are intelligible causes, and the others are sensible causes.

iii. 1; 996 a 4. And the most difficult and perplexing question of all is whether unity and being are not something different from things, as Plato and the Pythagoreans say, but their very essence, or whether the underlying substance is something different, such as Love, as Empedocles says, or as another says, fire, or water, or air.

iii. 4; 1001 a 9. Plato and the Pythagoreans assert that neither being Iior unity is something different from things, but that it is the very nature of them, as though essence itself consisted in unity and existence.

vii. 10; 1036 b 17. So it turns out that many things of which the forms appear different have one Form, as the Pythagoreans discovered; and one can say that there is one Form for everything, and hold that others are not forms, and thus all things will be one.

x. 2; 1053 b 11. Whether the One itself is a sort of essence, as first the Pythagoreans and later Plato affirmed...

xii. 7; 1072 b 31. And they are wrong who assume, as do the Pythagoreans and Speusippus, that the most beautiful and the best is not in the beginning of things, because the first principles of plants and animals are indeed causes, but that which is beautiful and perfect is in what comes from these first principles.

xiii. 4; 1078 b 21. The Pythagoreans [before Democritus] only defined a few things, the concepts of which they reduced to numbers, as for instance opportunity or justice or marriage...

xii. 6; 1080 b 16. The Pythagoreans say that there is but one number, the mathematical, but that things of sense are not separated from this, for they are composed of it; indeed, they construct the whole heaven out of numbers, but not out of abstract numbers, for they assume that the units have magnitude; but how the first unit was so constituted to have magnitude they seem at a loss to say.

xiii. 6; 1080 b 31. All, as many as regard the one as the element and first principle of things, except the Pythagoreans, assert that numbers are based on the unit; but the Pythagoreans assert, as has been remarked, that numbers have magnitude.

xiii. 8; 1083 b 9. The Pythagorean standpoint has on the one hand fewer difficulties than those that have been discussed, but it has new difficulties of its own. The fact that they do not regard number as separate removes many of the contradictions; but it is impossible that bodies should consist of numbers, and that this number should be mathematical. Nor is it true that indivisible elements have magnitude; but, granted that they have this quality of indivisibility, the units have no magnitude -- for how can magnitude be composed of indivisible elements? But arithmetical number consists of units. For these say that things are number; at least, they adapt their speculations to bodies as if they consist of numbers.

xiv. 3; 1090 a 20. On the other hand the Pythagoreans, because they see many qualities of numbers in bodies perceived by sense, regard objects as numbers, not as separate numbers, but as derived from numbers. And why? Because the qualities of numbers exist in a musical scale, in the heaven and in many other things. But for those who hold that number is mathematical only, it is impossible on the basis of their hypothesis to say any such thing, and it has already been remarked that there can be no science of these numbers. But we say, as above, that there is a science of numbers. Evidently the mathematical does not exist by itself, for in that case its qualities could not exist in bodies. In such a matter the Pythagoreans are restrained by nothing; when, however, they construct out of numbers physical bodies -- out of numbers that have neither weight nor lightness, bodies that have weight and lightness -- they seem to be speaking about another heaven and other bodies than those perceived by sense.

Nicomachean Ethics. i. 6; 1096 b 5. And the Pythagoreans seem to speak more persuasively about it, putting unity in the column of good things.

ii. 6; 1106 b 29. Evil partakes of the nature of the Unlimited, Good of the Limited, as the Pythagoreans conjectured.

v. 5; 1132 b 21. Reciprocity seems to some to be absolutely just, as the Pythagoreans say; for these defined the just as that which is reciprocal to another.

Moralia. i. 1; 1182 a 11. First Pythagoras attempted to speak concerning virtue, but he did not speak correctly; for bringing virtues into correspondence with numbers, he did not make any distinct.
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Re: The Pythagorean Sourcebook and Library

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THE WRITINGS of the doxographers, the Vetusta Placita of Aetius, the Placita Philosophorum attributed to Plutarch, and so on, ultimately derive from a work of Theophrastus, Physikon Doxon. "Opinions of the Natural Philosophers." In this work Theophrastus compiled information on the doctrines of different philosophers by subject, serving to compare and contrast them with one another.

The fragments of the doxographers were collected together and published by Hermann Diels as Doxographi Graeci, Berlin, 1879, and the corresponding numeration of Diels' edition is reproduced at the beginning of each fragment. These fragments relating to the Pythagoreans, like the preceding passages from Plato and Aristotle, are translated by Arthur Fairbanks and reproduced from his The First Philosophers of Greece: An Edition and Translation of the Remaining Fragments of the Pre-Sokratic Philosophers, Together with a Translation of the More Important Accounts of their Opinions Contained in the Early Epitomes of their Works, New York, Charles Scribner's Sons, 1898.


Aetius, Plac. i. 3; Dox. 280. And again from another starting-point, Pythagoras, son of Mnesarchus, a Samian, who was the first to call this matter by the name of philosophy, assumed as first principles the numbers and the symmetries existing in them, which he calls harmonies, and the elements compounded of both, that are called geometrical. And again he includes the Monad and the Indefinite Dyad among the first principles; and for him one of the first principles tends toward the creative and form-giving cause, which is intelligence, that is God, and the other tends toward the passive and material cause, which is the visible universe. And he says that the starting-point of Number is the Decad; for all Greeks and all barbarians count as far as ten, and when they get as far as this they return to the monad. And again, he says, the power of ten is in four and the tetrad. And the reason is this: if anyone from the monad adds the numbers in a series as far as four, he will fill out the number ten [i .e., 1 + 2 + 3 + 4 = 10], but if he goes beyond the number of the tetrad, he will exceed ten. Just as if one should add one and two and should add to these three and four, he will fill out the number ten; so that according to the monad number [actually] is in the ten, but potentially in the four. Wherefore the Pythagoreans were wont to speak as though the greatest oath were the Tetrad:

By him that transmitted to our soul the Tetraktys,
The spring and root of ever-flowing nature.

And our soul, he says, is composed of the Tetrad, for this is intelligence, understanding, opinion, and sense, from which things come every art and science, and we ourselves become reasoning beings. The Monad, however, is intelligence, for intelligence sees according to the Monad. As for example, men are made up of many parts, and part by part they are devoid of sense and comprehension and experience, yet we perceive that man as one alone, whom no being resembles, possessing these qualities; and we perceive that a horse is one, but part by part it is without experience. For these are forms and classes according to monads. Wherefore, assigning this limit with reference to each one of these, they speak of a reasoning being and a neighing being. On this account then the Monad is intelligence by which we perceive these things. And the Indefinite Dyad is fittingly science, for all proof and persuasion is part of science, and further every syllogism brings together what is questioned out of some things that are agreed upon, and easily proves something else; and science is the comprehension of these things, wherefore it would be the Dyad. And opinion as the result of comprehending them is fittingly the Triad, for opinion has to do with many things, and the Triad is quantity, as 'The thrice-blessed Danaoi.' On this account then he includes the Triad... And their sect is called Italic because Pythagoras taught in Italy, having left Samos, his fatherland, of dissatisfied with the tyranny of Polycrates.

i. 7; Dox. 302. Pythagoras held that one of the first principles, the Monad, is God and the Good, which is the origin of the One, and is itself Intelligence, but the Indefinite Dyad is a daimon and bad, surrounding which is the mass of matter.

i. 8; Dox. 307. Divine spirits (daimones) are psychical beings, and heroes are souls separated from bodies; good heroes are good souls, bad heroes are bad souls.

i. 9; Dox. 307. The followers of Thales and Pythagoras and the Stoics held that matter is variable and changeable and transformable and is in a state of flux, the whole through the whole.

i. 10; Dox. 309. Pythagoras asserted that the so-called forms and ideas exist in numbers and their harmonies, and in what are called geometrical objects, apart from bodies.

i. 11; Dox. 310. Pythagoras and Aristotle asserted that the first causes are immaterial, but that other causes involve a union or contact with material substance [so that the world is material].

i. 14; Dox. 312. The followers of Pythagoras held that the universe is a sphere according to the form of the four elements, but the highest, fire, alone is conical.

i. 15; Dox. 314. The Pythagoreans call color the manifestation of matter.

i. 16; Dox. 314. Bodies are subject to change of condition, and are divisible to infinity.

i. 18; Dox. 316. (After a quotation from Aristotle, Physics, iv. 4; 212 a 20.) And in his first book on the philosophy of Pythagoras he writes that the heaven is one, and that time and wind and the void which always defines the places of each thing, are introduced from the infinite. And among other things he says that place is the immovable Limit of what surrounds the world, or that in which bodies abide and are moved, and that it is full when it surrounds body on every side, and empty when it has absolutely nothing in itself. Accordingly it is necessary for place to exist, and body, and it is never empty except from the standpoint of thought, for the nature of it in perpetuity destroys the interrelation of things and the combination of bodies; motions arise according to the place of bodies that surround and oppose each other, and no infiniteness is lacking, either of quality or of extent.

i. 20; Dox. 318. Pythagoras said that time is the sphere which surrounds the world.

i. 21; Dox. 318. Pythagoras. Plato: Motion is a certain otherness or difference in matter.

i. 24; Dox. 320. Pythagoras, and all that assume matter is subject to change, assert that genesis and destruction in an absolute sense take place, for from change of the elements, modification and separation of them there takes place juxtaposition and mixture, and intermingling and melting together.

Aetius, Plac. ii. l; Dox. 327. Pythagoras first named the circumference of all things the kosmos by reason of the order in it.

ii. 4; Dox. 330. Pythagoras, Plato, and the Stoics held that the universe is brought into being by God. And it is perishable so far as its nature is concerned, for it is perceived by sense, and therefore material; it will not however be destroyed in accordance with the foreknowledge and plan of God.

ii. 6; Dox. 334. Pythagoras: The universe is made from five solid figures which are also called mathematical; of these he says that earth has arisen from the cube, fire from the pyramid, air from the octahedron, and water from the icosahedron, and the sphere of the All from the dodecahedron.

ii. 9; Dox. 338. The followers of Pythagoras hold that there is a void outside the universe into which the universe breathes forth, and from which it breathes in.

ii. 10; Dox. 339. Pythagoras, Plato, Aristotle: The right hand side of the universe is the eastern part from which comes the beginning of motion, and the left hand side is the west. They say the universe has neither height nor depth, in which statement height means distance from below upwards, and depth from above downwards. For none of the distance thus described exists for the universe, inasmuch as it is disposed around the middle of itself, from which it extends toward the All, and with reference to which it is the same on every side.

ii. 12; Dox. 340. Thales, Pythagoras, and their followers: The sphere of the whole heaven is divided into five circles which they call zones; the first of these is called the arctic zone and is ever visible, the second the summer solstice, the third the equinoctial, the fourth the winter solstice, and the fifth the antarctic zone, which is invisible. And the ecliptic called the zodiac in the three middle ones is projected to touch the three middle ones. And the meridian crosses all these from the north to the opposite quarter at right angles. It is said that Pythagoras was the first to recognize the slant of the zodiacal circle which Oenopides of Chios appropriated as his own discovery.

ii. 13; Dox. 343. Heracleides and the Pythagoreans asserted that each world of the stars is air and aether surrounding earth in the infinite aether. And these doctrines are brought out in the Orphic writings, for they [likewise] construct each world of the stars.

ii. 22; Dox. 352. The Pythagoreans: The sun is spherical.

ii. 23; Dox. 353. Plato, Pythagoras, Aristotle: The solstices lie along the slant of the zodiacal circle, through which the sun goes along the zodiac, and with the accompaniment of the tropical circles, and all these things the globe also shows.

ii. 24; Dox. 354. An eclipse takes place when the moon comes past.

ii. 25; Dox. 357. Pythagoras: The moon is a mirror-like body.

ii. 29; Dox. 360. Some of the Pythagoreans (according to the Aristotelian account and Philip of Opus) said that an eclipse of the moon takes place sometimes by the interposition of the earth, sometimes by the interposition of the counterearth (antichthon). But it seems to some more recent thinkers that it takes place by a spreading of the flame little by little as it is gradually kindled, until it gives the complete full moon, and again, in like manner, it grows less until the conjunction, when it is completely extinguished.

ii. 30; Dox. 361. Some of the Pythagoreans, among them Philolaus, said that the earthy appearance of the moon is due to its being inhabited by animals and by plants, like those on our earth, only greater and more beautiful; for the animals on it are fifteen times as powerful, not having any sort of excrement, and their day is fifteen times as long as ours. But others said that the outward appearance in the moon is a reflection on the other side of the inflamed circle of the sea that is on our earth.

ii. 32; Dox. 364. Some regard the greater the sixty year period, among whom are Oenopides and Pythagoras.

Aetius, Plac. iii. 1; Dox. 364. Some of the Pythagoreans said that the Milky Way is the burning of a star that fell from its own foundation, setting on fire the region through which it passed in a circle, as Phaethon was burned. And others say that the course of the sun arose in this manner at the first. And certain ones say that the appearance of the sun is like a mirror reflecting its rays toward the heaven, and therefore it happens at times to reflect its rays on the rainbow in the clouds.

iii. 2; Dox. 366. Some of the followers of Pythagoras say that a comet is one of the stars which are not always shining, but which emit their light periodically through a certain definite time; but others say that it is the reflection of our vision into the sun, like reflected images.

iii. 14; Dox. 378. Pythagoras: The earth, after the analogy of the sphere of the All, is divided into five zones, arctic, antarctic, summer, winter and equinoctial; of these the middle one he defines to be the middle of the earth, called for this very reason the torrid zone, the inhabited one [the one between the arctic and the torrid zones] being well-tempered...

Aetius, Plac. iv. 2; Dox. 386. Pythagoras holds that number moves itself, and he takes number as an equivalent for intelligence.

iv. 4, Dox. 389. Pythagoras, Plato: According to a superficial account the soul is of two parts, the one possessing, the other lacking, reason; but according to close and exact examination, of three parts, for the unreasoning part they divide into the emotion and the desires.

Theodor. v. 20; Dox. 390. The successors of Pythagoras saying that the body is a mixture of five elements (for they ranked aether as a fifth along with the four), held that the powers of the soul are of the same number as these. And these they named intelligence and wisdom and understanding and opinion and sense-perception.

Aetius, Plac. iv. 5; Dox. 391. Pythagoras: The principle of life is about the heart, but the principle of reason and intelligence is about the head.

iv. 5; Dox. 392. Pythagoras, et al.: The intelligence enters from without.

iv. 7; Dox. 392. Pythagoras, Plato: The soul is imperishable.

iv. 9; Dox. 396. Pythagoras, et al.: The sense-perceptions are deceptive.

iv. 9; Dox. 397. Pythagoras, Plato: Each of the sensations is pure, proceeding from each single element. With reference to vision, it is of the nature of aether; hearing, of the nature of wind; smell, of the nature of fire; taste, of the nature of moisture; touch, of the nature of earth.

iv. 14; Dox. 405. The followers of Pythagoras and of the mathematicians on reflections of vision: For vision moves directly as it were against the bronze [of a mirror], and meeting with a firm, smooth surface, it is turned and bent back on itself, meeting some such experience as when the arm is extended and then bent back to the shoulder.

iv. 20; Dox. 409. Pythagoras, Plato, Aristotle: Sound is immaterial. For it is not air, but it is the form about the air and the appearance (epiphaneia) after some sort of percussion which becomes sound; and every appearance is immaterial, for it moves with bodies, but is itself absolutely immaterial, as in the case of a bent rod the surface appearance suffers no change, but the matter is what is bent.

Aetius, Plac. v. 1; Dox. 415. Pythagoras did not admit the sacrificial part alone (of augury).

v. 3; Dox. 417. Pythagoras: Sperm is foam of the best part of the blood, a secretion from the nourishment, like blood and marrow.

v. 4; Dox. 417. Pythagoras, Plato, Aristotle: The power of seed is immaterial, like intelligence, the moving power; but the matter that is poured forth is material.

v. 20; Dox. 432. Pythagoras, Plato: The souls of animals called unreasoning are reasonable, not however with active reasoning powers, because of an imperfect mixture of the bodies and because they do not have the power of speech, as is the case of apes and dogs, for these have intelligence but not the power of speech.

Ar. Did. Ep. Fr. 32; Dox. 467. Apollodorus in the second book Concerning the Gods: It is the Pythagorean opinion that the morning and the evening star are the same.

Theophr. Phys. Op. Fr. 17; Dox. 492. Favorinus says that he [Pythagoras] was the first to call the heavens a kosmos and the earth spherical.

Cic., de Deor. Nat., i. 11; Philod., Piet. Fr. c 4 b; Dox. 533. For Pythagoras, who held that soul is extended through all the nature of things and mingled with them, and that from this our souls are taken, did not see that God would be separated and torn apart by the separation of human souls; and when souls are wretched, as might happen to many, then part of God would be wretched -- a thing which could not happen.

Hippol., Phil. 2; Dox. 555. There is a second philosophy not far distant from the same time, of which Pythagoras, whom some call a Samian, was the first representative. And this they call the Italian philosophy because Pythagoras fled the rule of Polycrates over the Samians and settled in an Italian city where he spent his life. The successive leaders of this sect shared the same spirit. And he in his studies of nature mingled astronomy and geometry and music [and arithmetic]. And thus he asserted that God is a monad, and examining the nature of number with especial care, he said that the kosmos produces melody and is put together with harmony, and he first proved the motion of the seven stars to be rhythm and melody. And in wonder at the structure of the universe, he decreed that at first his disciples should be silent, as if they were mystae who were coming into the order of the All; then when he thought they had sufficient education in the principles of truth, and had sought wisdom sufficiently in regard to stars and in regard to nature, he pronounced them pure and then bade them to speak. He separated his disciples into two groups, and called one esoteric, and the other exoteric. To the former he entrusted the more perfect sciences, to the latter the more moderate. And he dealt with magic, as they say, and himself discovered the art of physiognomy. Postulating both numbers and measures he was wont to say that the first principle of arithmetic embraced philosophy by combination, after the following manner:

Number is the first principle, a thing which is undefined, incomprehensible, having in itself all numbers which could reach infinity in amount. And the first principle of numbers is in substance the first Monad, which is a male monad, begetting as a father all other numbers. Secondly the Dyad is a female number, and the same is called by the arithmeticians even. Thirdly the Triad is a male number; this the arithmeticians have been wont to call odd. Finally, the Tetrad is a female number, and the same is called even because it is female.

All numbers, then, taken by classes are four -- but number is undefined in reference to class -- of which is composed the perfect number, the Decad. For the series one, two, three and four becomes ten, and its own name is kept in its essence by each of the numbers. Pythagoras said that this sacred Tetraktys is "the spring having the roots of ever-flowing nature" in itself, and from this numbers have their first principle. For the eleven and the twelve and the rest derive from the ten the first principle of their being. The four parts of the Decad, this perfect number, are called number, monad, power and cube. And the interweavings and minglings of these in the origin of growth are what naturally completes nascent number; for when a power is multiplied upon itself, it is the power of a power; and when a power is multiplied on a cube, it is the power of a cube; and when a cube is multiplied on a cube, the cube of a cube; thus all numbers, from which arise the genesis of what arises, are seven: number, monad, power, cube, power of a power, power of a cube, and cube of a cube.

He said that the soul is immortal, and that it changes from one body to another; so he was wont to say that he himself had been born before the Trojan war as Aethalides, and at the time of the Trojan war as Euphorbus, and after that as Hermontimus of Samos, then as Pyrrhos of Delos, fifth as Pythagoras. And Diodorus of Eretria and Aristoxenus the musician say that Pythagoras had come unto Zaratas of Chaldaea [i.e., Zoroaster]; and he set forth that in his view there were from the beginning two causes of things: father and mother. The father is light and the mother darkness; and the parts of light are warm, dry, light, swift; and of darkness are cold, moist, heavy, slow; and of all these the universe is composed, of male and female. And he says that the universe exists in accordance with musical harmony, so the sun also makes an harmonious period. And concerning the things that arise from the earth and the universe they say Zaratas spoke as follows: There are two divinities, one of the heavens and the other of the earth; the one of the earth produces things from the earth, and it is water; and the divinity of the heavens is fire with a portion of air, warm, and cold; wherefore he says that none of these things will destroy or even pollute the soul, for these are the essence of all things. And it is said that Zaratas forbade men to eat beans because he said that at the beginning and composition of all things when the earth was still a whole, the bean arose. And he says that the proof of this is that if one chews a bean to a pulp and exposes it to the sun for a certain time (for the sun will affect it quickly), it gives off the odor of human seed. And he says that there is another and clearer proof: if when a bean is in flower we were to take the bean and its flower, and putting it into a pitcher moisten it and then bury it in the earth, and after a few days dig it up again, we should see in the first place that it had the form of a womb, and examining it closely we should find the head of a child growing with it.

Pythagoras perished in a conflagration with his disciples in Croton in Italy. And it was the custom when one became a disciple to burn one's property and to leave one's money under a seal with Pythagoras, and one remained in silence sometimes three years, and sometimes five years, and studied. And immediately on being released from this one mingled with the others and continued as a disciple and made one's home with them; otherwise one took one's money and was sent off. The esoteric class were called Pythagoreans, and the others Pythagoristians. And those of the disciples who escaped the conflagration were Lysis and Archippus and Zalmoxis the slave of Pythagoras, who is said to have taught the Pythagorean philosophy to the Druids among the Celts. It is said that Pythagoras learned numbers and measures from the Egyptians. Astonished at the wisdom of the priests, which was deserving of belief and full of fancies and difficult to grasp, he imitated it and himself also taught his disciples to be silent, and obliged the student to remain quietly in rooms underneath the earth.

Epiph. Pro. i; Dox. 587. Pythagoras laid down the doctrine of the Monad and of foreknowledge and the prohibition on sacrificing to the Gods then believed in, and he bade men not to partake of beings that had life, and to refrain from wine. And he drew a line between the things from the moon upwards, calling these immortal, and those below, which he called mortal; and he taught the transmigration of souls from bodies into bodies even as far as animals and beasts. And he used to teach his followers to observe silence for a period of five years. Finally he named himself a God. [1]

Epiph. Haer. iii. 8; Dox. 390. Pythagoras the Samian, son of Mnesarchus, said that the Monad is God, and that nothing has been brought into being apart from this. He was wont to say that wise men ought not to sacrifice animals to the Gods, nor yet to eat what had life, nor beans, nor to drink wine. And he was wont to say that all things from the moon downward were subject to change, while from the moon upward they were not. And he said that the soul goes at death into other animals. And he bade his disciples to keep silence for a period of five years, and finally he named himself a God.

Herm. I.G.P. 16; Dox. 655. Others then from the ancient tribe, Pythagoras and his fellow tribesmen, revered and taciturn, transmitted other dogmas to me as mysteries, and this is the great and unspeakable ipse dixit: the Monad is the first principle of all things. From its forms and from numbers the elements arose. And he declared that the number, form and measure of each of these is somehow as follows: Fire is composed of twenty-four right-angled triangles, surrounded by four equilaterals. And each equilateral consists of six right- angled triangles, whence they compare it to the pyramid. Air is composed of forty-eight triangles, surrounded by eight equilaterals. And it is compared to the octahedron, which is surrounded by eight equilateral triangles, each of which is separated into six right-angled triangles so as to become forty-eight in all. And water is composed of one hundred and twenty triangles, surrounded by twenty equilaterals, and it is compared to the icosahedron, which is composed of one hundred and twenty equilateral triangle. And aither is composed of twelve equilateral pentagons, and is like a dodecahedron. And earth is composed of forty-eight triangles, and is surrounded by six equilateral tetragons, and it is like a cube. For the cube is surrounded by six tetragons, each of which is separated into eight triangles, so that they become in all forty-eight.

FIGURE 16. THE REGULAR SOLIDS. The regular solids, also known as the Platonic solids, were first described by Plato in his Timaeus. Plato identified the dodecahedron with the cosmic sphere (later identified with aither), and the four other solids with the four elements. Each one of the elemental "molecules" is constructed out of the triangular "atoms" shown below. The five regular solids are the only polyhedra that can be constructed out of the same regular polygons. The archetypal ratios and geometries with which they are associated underlie the structure and divisions of three-dimensional space.

DODECAHEDRON: Aither / 12 Sides
TETRAHEDRON: Fire / 4 Sides
CUBE: Earth / 6 Sides
OCTAHEDRON: Air / 8 Sides
ICOSAHEDRON : Water / 20 Sides




1. While Pythagoras may have thought of himself as having some type of special relationship with the God Apollo, there is no reason to believe that he ever thought of himself as being a God. In fact, other church fathers attributed to Pythagoras a statement which challenged any man who thought himself a God to create a universe.
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:11 am



THE IMPORTANCE OF THE QUATERNARY obtained by addition (that is to say 1 + 2 + 3 + 4) is great in music because all the consonances are found in it. But it is not only for this reason that all Pythagoreans hold it in highest esteem: it is also because it seems to outline the entire nature of the universe. It is for this reason that the formula of their oath was: "I swear by the one who has bestowed the Tetraktys to the coming generations, source of eternal nature, into our souls." The one who bestowed it was Pythagoras, and it has been said that the Tetraktys appears indeed to have been discovered by him.

The first quaternary is the one of which we've just spoken: it is formed by addition of the first four numbers.

The second is formed by multiplication, of even and odd numbers, starting from unity. Of these numbers, unity is the first because, as we have said, it is the principle of all the even numbers, the odd numbers and of all the odd-even numbers, and its essence is simple. Next comes three numbers from the odd as well as the even series. They allow for the unification of odd and even because numbers are not only odd or even. For this reason, in multiplication, two quaternaries are taken, one even, the other odd; the even in double ratio, the first of the even numbers being 2 which comes from unity doubled; the odd in triple ratio, the first of the odd numbers being the number 3 which arise from unity being tripled, so that unity is odd and even simultaneously and belongs to both. The second number in the even and double [series] is 2 and in the odd and triple is 3. The third of the order of even numbers is 4, and in the odd series, 9. The fourth among the even numbers is 8, and among the odd numbers, 27.


The ratios of the most perfect consonances are found in these numbers; even the tone is included. However unity contains the principle of ratio, of limit and of point. The second numbers, 2 and 3, have the side ratio, being prime, incomposite numbers, and measured only by the unit, and are consequently linear numbers. The third terms, 4 and 9, have the power of the squared surface, being equally equal (that is to say square numbers). The fourth terms, 8 and 27, have the power of the cubic solid, being equally equal equally (that is to say, cubic numbers). In this way, by virtue of the numbers from this tetraktys, growth proceeds from the limit and the point up to the solid. In fact, after the limit and the point comes the side, then the surface and finally the solid. It is with these numbers that Plato, in the Timaeus, constitutes the [world] soul. [1] The last of these seven numbers is equal to the sum of all the preceding, as we have 1+2+3+4+8+9=27.

There are then two quaternaries of numbers, one which is made by addition, the other by multiplication; and these quaternaries encompass the musical, geometric and arithmetic ratios of which the harmony of the universe is composed.

The third quaternary is that which, following the same proportion, embraces the nature of all magnitudes, for the place taken by unity, in the preceding quaternary, is that of the point in this one; and that of the numbers 2 and 3, having lateral (or linear) power, is here that of the line, through its double form, straight or circular, the straight line corresponding to the even number because it terminates at two points (the line and circle are given as examples here), and the circular to the odd, because it is composed of a single line without terminus.

And what, in the preceding quaternary, are the numbers 4 and 9, having the power of the surface, the two types of surface, the planar and the curved, are so (surface) in this one. Finally, what, in the preceding are the numbers 8 and 27, which have the power of the cube and of which one is even and the other odd, is constituted by the solid in this one. There are two kinds of solids, one with a curved surface, like the sphere or the cylinder, the other with a plane surface, such as the cube and the pyramid. This is the third tetraktys then, the one having the property of constituting any magnitude, through the point, the line, the surface and the solid.

The fourth quaternary is that of the simple bodies, fire, air, water and earth, and it offers the same proportion as the quaternary of numbers. The place occupied by unity in the quaternary of numbers is taken by fire in this one, air corresponds to the number 2, water to the number 3, earth to the number 4; such is indeed the nature of the elements according to their fineness or density, in such a way that fire is to air as 1 is to 2, to water as 1 is to 3, and to earth as 1 is to 4. The other relationships are also equal (that is to say, that air is to water as 2 is to 3, and so forth for the others).

The fifth quaternary is that of the shapes of simple bodies, for the pyramid is the figure of fire, the octahedron the figure of air, the icosahedron the figure of water and the cube the figure of earth.

The sixth is that of the created things, the seed being analogous to unity and the point. A growth in length is analogous to the number 2 and the line, and a growth in width is analogous to the number 3 and to the surface, and finally a growth in thickness is analogous to the number 4 and to the solid.

The seventh quaternary is that of societies. Man is principle and is thus unity. The family corresponds to the number 2, the village to the number 3 and the city to the number 4; for these are the elements which comprise the nation.

All of these quaternaries are material and perceptible.

The eighth contains faculties by which we are able to form judgment on the preceding, and which are its intellectual part, namely: thought, science, opinion and feeling. And certainly thought, in its essence, must be assimilated to unity; science is the number 2, because it is the science of all things; opinion is like the number 3, because it is something between science and ignorance; and finally feeling is like the number 4 because it is quadruple, the sense of touch being common to all, all the senses being motivated through contact.

The ninth quaternary is that which composes the living things, body and soul, the soul having three parts, the rational, the emotional and the willful; the fourth part is the body in which the soul resides.

The tenth quaternary is that of the of the seasons of the year, through the succession of which all things take birth, that is, spring, summer, autumn and winter.

The eleventh is that of the ages: childhood, adolescence, maturity and old age.

There are thus eleven quaternaries. The first is that of the numbers which are formed by addition, the second is that of the numbers formed by multiplication, the fourth is that of magnitudes, the fifth is that of simple bodies, the sixth is that of created things, the seventh is that of societies, the eighth is that of the faculties of judgment, the ninth is that of the living things, the tenth is that of the seasons, and the eleventh is that of the ages. They are proportional to one another, since what is unity in the first and the second quaternary, the point is in the third, fire is in fourth, the pyramid in the fifth, the seed in the sixth, man in the seventh, thought in the eighth, and so forth with the others following the same proportion.

Thus the first quaternary is 1, 2, 3, 4. The second is unity, the side, the square, the cube. The third is the point, the line, the surface, the solid. The fourth is fire, air, water, earth. The fifth is the pyramid, the octahedron, the icosahedron, the cube. The sixth is the seed, the length, the width, the height. The seventh is man, the family, the village, the city. The eighth is thought, science, opinion, sense. The ninth is the rational, the emotional and the willful parts of the soul, and the body. The tenth is spring, summer, autumn, and winter. The eleventh is childhood, adolescence, maturity and old age. And the perfect world which results from these quaternaries is geometrically, harmonically and arithmetically arranged, containing in power the entire nature of number, every magnitude and every body, whether simple or composite. It is perfect because everything is part of it, and it is itself a part of nothing else. This is why the Pythagoreans used the oath whose formula we have reported, and through which all things are assimilated to number.

From Theon of Smyrna: Mathematics Useful for Understanding Plato, Chapter 38. Translated by Robert and Deborah Lawlor. San Diego, Wizards Bookshelf, 1979. Reproduced with permission of the publisher.



1. Plato, Timaeus 36BC.
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:20 am




PLUTARCH AND PLOTINUS inform us that the Pythagoreans called the One Apollo because of its lack of multiplicity -- this is both a clever pun and a revealing statement, for a-pollon in Greek means literally "not of many."

Such, then, was the Greek style of "theological arithmetic." This form of number symbolism became quite popular in late antiquity and much of it was transmitted by Christian writers through Medieval times. The symbolism finds its basis in the Pythagorean observation that the primary numbers represent far more than quantitative signs: each one of the primary numbers is a qualitative, archetypal essence, possessing a distinct, living personality. This personality can be directly intuited by studying the archetypal manifestations of these principles in the realms arithmetic (number in itself), geometry (number in space) and harmonics (number in time).

The following list represents a compilation of the titles from the anonymous Theology of Arithmetic which was based closely on a work by Iamblichus. Since the first printing of this volume, however, a complete, fully annotated translation of The Theology of Arithmetic has appeared, which represents the first translation of this work into any European language (The Theology of Arithmetic, translated by Robin Waterfield, Phanes Press, Grand Rapids, 1988.) In the text, explanations are given for the various appellations. For more on arithmology also see book three of Thomas Taylor's Theoretic Arithmetic of the Pythagoreans and the section on number symbolism in Thomas Stanley's Pythagoras.


Instrument of Truth
A Chariot
Immutable Truth and Invulnerable Destiny
A Seed
Fabricator (demiurge)
True Happiness (eudaimonia)
The Equality in Increase and |Decrease
A Ship
Essence (ousia)
The Innkeeper (pandokeus), "that which takes in all"
The Pattern or Model (paradeigma)
The Moulder
The First (Proteus)
Harmony (symphonia)
Order (taxis)
A Friend
Infinite Expanse (chaos)


Indefinite (aoristos)
The Unlimited (apeiron)
Without Form or Figure
The Each of Two
Falling Short, Defect
The Ratio (logos) in Proportion (analogia)
The Thing with Another
Rhea (the wife of Kronos, but also "flow")
That Which is To Be Endured; Misery, Distress
Boldness, Audacity (tolma)


Proportion (analogia)
Knowledge (gnosis)
Every Thing
Good Counsel
The Mean Between Two Extremes
Oneness of Mind
The All


Nature of Change
Holding the Key of Nature


Lack of Strife
Boubastia (named after the Egyptian divinity Boubastis)
Manifesting Justice


Resembling Justice
The Thunder-Stone
Amphitrite (Poseidon's wife; a verbal pun: on both sides [amphis] three [trias])
Finest of All
In Two Measures
Form of Forms
Far-Shooting (name of Apollo)
Possessing Wholeness
Cure-All (panacea)


The Forager (epithet of Athena)
Citadel (akropolis)
Hard to Subdue Defence
Due Measure (kairos)
Virgin (parthenos)
Revered Seven (septas + sebomai = heptas)
Bringing to Completion
Fortune, Fate


Untimely Born
Seat or Abode
All Harmonious


Brother and Consort of Zeus
Absence of Strife
Far-Working (epithet of Apollo)
Maiden (kore)
Of the Kouretes
Oneness of Mind
Horizon (because it limits the series of units before returning to the Decad)
Crossing or Passage
Consort and Brother
Bringing to Perfection (Telesphorus)


Eternity (aeon)
All Perfect
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:22 am

FIGURE 18. THE DIVINE MONOCHORD. This particular monochord is tuned in the key of G, while the examples [below] and in the introduction use the key of C. The three top notes on this monochord are incorrectly placed.



AS NOTED in the introductory essay, the structure of the musical scale possesses a great deal of significance in Pythagorean thought as it is an excellent example of the principle of mathematical harmonia at work. In the case of the scale, the "opposites" of the high (2) and the low (1) -- the two extremes of the octave -- are united in one continuum of tonal relationships through the use of a variety of forms of proportion which actively mediate between these two extremes.

The best way to understand the mathematical principles of harmonic mediation involves actually charting out and playing out the ratios of the scale on the monochord. In constructing a monochord, it is best to make it as long as possible, perhaps in the region of 4-5 feet, as that makes it easier to differentiate between the harmonic nodal points at the high end of the spectrum (see fig. 6).

It is useful at first to play out the harmonic overtone series. Measure the exact length of the string and then mark off the overtone intervals: 1/2 the string length, 1/3 the string length, 1/4 the string length, etc. It is possible to play out the overtone series without the use of the bridge; simply pluck the string about 1 inch from either end while simultaneously touching the nodal point with the other hand. It will be noted that there is an inverse relationship between the vibrational frequency of the tone and the string length. This is also illustrated in the above chart: hence a tone with a vibration of 2 is associated with a string division of .5 or 1/2. It is also useful at this point to play out the harmonic "Tetraktys," or the perfect consonances: 1:2 (octave), 2:3 (perfect fifth), and 3:4 (perfect fourth). Listen carefully to these ratios and reflect on the fact that you are actually hearing the relationships between these primary whole numbers.

To "tune" the monochord to the ratios of the Pythagorean scale use the string length ratios in the above chart, multiplying these ratios by the length of the string. Mark off these intervals, along with the corresponding notes, on the sounding board as they are carefully measured out.

Having marked out the Pythagorean scale, it might be useful at this point to review the material in the introductory essay relating to the harmonic proportion and then to play out these relations:

1) Play out the relationship of the octave (1:2). These are the two tonal extremes which must be united.

2) Play out the arithmetic mean linking together the extremes: C-G-c, or 6-9-12. This is the perfect fifth, the strongest musical relationship (2:3).

3) Play out the harmonic mean linking together the two extremes: C-F-c, or 6-8-12. This is the perfect fourth, the next strongest musical relationship (3:4).

4) Now play out the harmonic or musical proportion which is the basis of the musical scale: C-F-G-c, or 6:8::9::12. Play this out as a continued proportion and then the individual parts. Play out the two perfect fifths 6:9 and 8:12. Play out the two perfect fourths 6:8 and 9:12. Then play out the whole tone 8:9.

5) Having played out the harmonic foundation of the scale, now "fill in" the remaining 8:9 whole tone intervals. Play out C-D, D-E, G-A, and A-B. Along with F-G, these are all in the 8:9 ratio.

6) Play out the ratio of the leimma or the semitone: E-F and B-c. The leimma is the relationship between the perfect fourth and three whole tones.

7) Finally play out the entire scale: C-D-E-F-G-A-B-c. Through the use of arithmetic, harmonic and geometric proportion the two extremes have been successfully united.
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:28 am



NOT ONLY did the early Pythagoreans make many contributions in the realm of philosophy, but their mathematical studies laid the foundation for the development of Greek geometry, and many portions of Euclid's Elements can be traced back to mathematical discoveries of the Pythagorean school.

This listing of early Pythagorean mathematical discoveries is excerpted from Thomas Heath's History of Greek Mathematics, vol. I, pp, 166-169.


1. They were acquainted with the properties of parallel lines, which they used for the purpose of establishing by a general proof the proposition that the sum of the three angles of any triangle is equal to two right angles. This latter proposition they again used to establish the well-known theorems about The sums of the exterior and interior angles, respectively, of any polygon.

2. They originated the subject of equivalent areas, the transformation of an area of one form into another of different form and, in particular, the whole method of application of areas, constituting a geometrical algebra, whereby they effect the equivalent of the algebraical processes of addition, subtraction, multiplication, division, squaring, extraction of the square root, and finally the complete solution to the mixed quadratic equation x2 ± pxq = 0 so far as its roots are real. Expressed in terms of Euclid, this means the whole content of Book I. 35-48 and Book II. The method of application of areas is one of the most fundamental in the whole of later Greek geometry; it takes its place by the side of the powerful method of proportion; moreover, it is the starting point of Apollonius' theory of conics, and the three fundamental terms, parabole, ellipsis, and hyperbole used to describe the three separate problems in 'application' were actually employed by Apollonius to denote the three conics, names which, of course, are those which we use to-day. Nor was the use of the geometrical algebra for solving numerical problems unknown to the Pythagoreans; this is proved by the fact that the theorems of Eucl. II. 9, 10 were invented for the purpose of finding successive integral solutions of the indeterminate equations

2x2 - y2 = ± 1

3. They had a theory of proportion pretty fully developed. We know nothing of the form in which it was expounded; all we know is that it took no account of incommensurable magnitudes. Hence we conclude that it was a numerical theory, a theory on the same lines as that contained in Book VII of Euclid's Elements.

They were aware of the properties of similar figures. This is clear from the fact that they must be assumed to have solved the problem, which was, according to Plutarch, attributed to Pythagoras himself, of describing a figure which shall be similar to one given figure and equal in area to another given figure. This implies a knowledge of the proposition that similar figures (triangles or polygons) are to one another in the duplicate ratio of corresponding sides (Eucl. VI. 19, 20). As the problem is solved in Eucl. VI. 25, we assume that, subject to the qualification that their theorems about similarity, &c., were only established of figures in which corresponding elements are commensurable, they had theorems corresponding to a great part of Eucl., Book VI.

Again, they knew how to cut a straight line in extreme and mean ratio (Eucl. VI. 30); [1] this problem was presumably solved by the method used in Eucl. II. 11, rather than by that of Eucl. VI. 30, which depends on the solution of a problem in the application of areas more general than the methods of Book II enable us to solve, the problem namely of Eucl. VI. 29.

4. They had discovered, or were aware of the existence of, the five regular solids. These they may have constructed empirically by putting together squares, equilateral triangles, and pentagons. This implies that they could construct a regular pentagon and, as this construction depends upon the construction of an isosceles triangle in which each of the base angles is double of the vertical angle, and this again on the cutting of a line in extreme and mean ratio, we may fairly assume that this was the way in which the construction of the regular pentagon was actually evolved. It would follow that the solution of problems by analysis was already practised by the Pythagoreans, notwithstanding that the discovery of the analytical method is attributed by Proclus to Plato. As the particular construction is practically given in Eucl. IV. 10, 11, we may assume that the content of Eucl. IV was also partly Pythagorean.

5. They discovered the existence of the irrational in the sense that they proved the incommensurability of the diagonal of a square with reference to its side; in other words, they proved the irrationality of √2. At; a proof of this is referred to by Aristotle in terms which correspond to the method used in a proposition interpolated in Euclid, Book X, we may conclude that this proof is ancient, and therefore that it was probably the proof used by the discoverers of the proposition. The method is to prove that, if the diagonal of a square is commensurable with the side, then the same number must be both odd and even; here then we probably have of early Pythagorean use of the method of reductio ad absurdum.

Not only did the Pythagoreans discover the irrationality of √2; they showed, as we have seen, how to approximate as closely as we please to its numerical value. After the discovery of this one case of irrationality, it would be obvious that propositions theretofore proven by means of the numerical theory of proportion, which was inapplicable to incommensurable magnitudes, were only partially proved. Accordingly, pending the discovery of a theory of proportion applicable to incommensurable as well as commensurable magnitudes, there would be an inducement to substitute, where possible, for proofs employing the theory of proportion other proofs independent of that theory. This substitution is carried rather far in Euclid, Books I-IV; it does not follow that the Pythagoreans remodelled their proofs to the same extent as Euclid felt bound to do.



1. The extreme and mean division of the line is an important mathematical and geometrical ratio which underlies various universal forms. This is the so-called "Divine Proportion," "Golden Section," or Phi ratio. On the properties and significance of this principle see Ghyka, The Geometry of Art and Life, and other titles on sacred geometry listed in the bibliography.
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Re: The Pythagorean Sourcebook and Library

Postby admin » Wed Nov 13, 2013 5:29 am


Analogia -- Literally, "through proportion." Hence, continued geometrical proportion or ratio. See logos.

Apeiron -- Boundless; Unlimited; Infinite; Indefinite. One of the Pythagorean first principles in the Table of Opposites. See peras and Indefinite Dyad.

Harmonia -- A "joint" or "fitting together;" hence, the musical scale comprised within the octave. Also, music per se; philosophically, the principle of Union, opposite Strife. Also, a Pythagorean name of the number Three, because a third element must be introduced to mediate between or join together two separate principles or numerical values.

Indefinite Dyad -- Plato's term for the Pythagorean principle of Apeiron, as contrasted with the One, the principle of Limit.

Kosmos -- "Order." Also, "ornament." First applied to the universe by Pythagoras, hence cosmos means world-order. Also, a Pythagorean name of the numbers Six and Ten.

Logos -- Usually translated "Word" or "Reason." In the mathematical and Pythagorean sense, the same as the Latin ratio, i.e. "proportion;" hence also, a principle of mediation. Can also mean "principle;" the plural logoi can be translated as "principles," "reasons" or "causes," or (mathematical) "ratios."

Mean or median -- The middle term in a mathematical proportion which links two extreme terms together in harmonia. The three most important are the Arithmetic, Harmonic, and Geometric means, which underlie the structure of the musical scale in Pythagorean tuning. In the following equations, the two extremes are A and C, and the mean term is B.

Arithmetic Mean: B = (A +C) / 2
Harmonic Mean: B = 2AC / A + C
Geometric Mean: B = √A x C

Monochord (kanon) -- A one-stringed musical instrument with a movable bridge used for dividing the string at any length. The monochord is used to demonstrate the harmonic overtone series and the principles on which the musical scale is based.

Peras -- The principle of Limit or Boundary. The opposite of Apeiron, the Unlimited.

Symphonia -- Literally, "sounding together." "Harmony," agreement or concord. The term applies to the perfect intervals or consonance of the octave, fifth, and fourth, par excellence. This is the modern meaning of the word "harmony," but not the ancient one.

Tetraktys -- (from tetras, four). "Fourness." Also, the first four numbers, especially when arranged in an equilateral triangle, the sum of which is the number Ten. Hence also, the Decad.

The Tetraktys symbolizes the perfection of Number and the elements which comprise it. The Tetraktys also contains the symphonic ratios which make possible the musical scale, i.e., 1:2, the octave; 2:3, the perfect fifth; and 3:4, the perfect fourth.
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