(2) All things accord in number:

which he very frequently uttered to all his disciples. Or again, Friendship is equality; equality is friendship . Or in the word cosmos , i. e. the world ; or by Jupiter, in the word philosophy , or in the so much celebrated word tetractys . All these and many other inventions of the like kind, were devised by Pythagoras for the benefit and amendment of his associates; and they were considered by those that understood them to be so venerable, and so much the progeny of divine inspiration, that the following was adopted as an oath by those that dwelt together in the common auditory:

(99) "Perfect numbers, therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be. And evil indeed is opposed to evil, but both are opposed to one good. Good, however, is never opposed to good, but to two evils at one and the same time. Thus timidity is opposed to audacity, to both [of] which the want of true courage is common; but both timidity and audacity are opposed to fortitude. Craft also is opposed to fatuity, to both [of] which the want of intellect is common; and both these are opposed to prudence. Thus, too, profusion is opposed to avarice, to both [of] which illiberality is common; and both these are opposed to liberality. And in a similar manner in the other virtues; by all [of] which it is evident that perfect numbers have a great similitude to the virtues. But they also resemble the virtues on another account; for they are rarely found, as being few, and they are generated in a very constant order. On the contrary, an infinite multitude of superabundant and diminished numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite."

(7) Now if we do not mean anything by Relation but are victims of words, none of the relations mentioned can exist: Relation will be a notion void of content.

Suppose however that we do possess ourselves of objective truth when in comparing two points of time we pronounce one prior, or posterior, to the other, that priority does entail something distinct from the objects to which it refers; admit an objective truth behind the relation of left and right: does this apply also to magnitudes, and is the relation exhibiting excess and deficiency also something distinct from the quantities involved?

Now one thing is double of another quite apart from our speech or thought; one thing possesses and another is possessed before we notice the fact; equals do not await our comparison but- and this applies to Quality as well as Quantity- rest upon an identity existing between the objects compared: in all the conditions in which we assert Relation the mutual relation exists over and above the objects; we perceive it as already existent; our knowledge is directed upon a thing, there to be known- a clear testimony to the reality of Relation.

In these circumstances we can no longer put the question of its existence. We have simply to distinguish: sometimes the relation subsists while the objects remain unaltered and even apart; sometimes it depends upon their combination; sometimes, while they remain unchanged, the relation utterly ceases, or, as happens with right and near, becomes different. These are the facts which chiefly account for the notion that Relation has no reality in such circumstances.

Our task, thus, is to give full value to this elusive character of Relation, and, then to enquire what there is that is constant in all these particular cases and whether this constant is generic or accidental; and having found this constant, we must discover what sort of actuality it possesses.

It need hardly be said that we are not to affirm Relation where one thing is simply an attribute of another, as a habit is an attribute of a soul or of a body; it is not Relation when a soul belongs to this individual or dwells in that body. Relation enters only when the actuality of the relationships is derived from no other source than Relation itself; the actuality must be, not that which is characteristic of the substances in question, but that which is specifically called relative. Thus double with its correlative, half gives actuality neither to two yards' length or the number two, nor to one yard's length or the number one; what happens is that, when these quantities are viewed in their relation, they are found to be not merely two and one respectively, but to produce the assertion and to exhibit the fact of standing one to the other in the condition of double and half. Out of the objects in a certain conjunction this condition of being double and half has issued as something distinct from either; double and half have emerged as correlatives, and their being is precisely this of mutual dependence; the double exists by its superiority over the half, and the half by its inferiority; there is no priority to distinguish double from half; they arise simultaneously.

It is another question whether they endure simultaneously. Take the case of father and son, and such relationships; the father dies, but the other is still his son, and so with brothers. Moreover, we see likeness where one of the like people is dead.

(5) Such, then, is the style of the example in arithmetic. And let the testimony of geometry be the tabernacle that was constructed, and the ark that was fashioned, - constructed in most regular proportions, and through divine ideas, by the gift of understanding, which leads us from things of sense to intellectual objects, or rather from these to holy things, and to the holy of holies. For the squares of wood indicate that the square form, producing fight angles, pervades all, and points out security. And the length of the structure was three hundred cubits, and the breadth fifty, and the height thirty; and above, the ark ends in a cubit, narrowing to a cubit from the broad base like a pyramid, the symbol of those who are purified and tested by fire. And this geometrical proportion has a place, for the transport of those holy abodes, whose differences are indicated by the differences of the numbers set down below.

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

(11) To Pythagoras music was one of the dependencies of the divine science of mathematics, and its harmonies were inflexibly controlled by mathematical proportions. The Pythagoreans averred that mathematics demonstrated the exact method by which the good established and maintained its universe. Number therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. After discovering these harmonic ratios, Pythagoras gradually initiated his disciples into this, the supreme arcanum of his Mysteries. He divided the multitudinous parts of creation into a vast number of planes or spheres, to each of which he assigned a tone, a harmonic interval, a number, a name, a color, and a form. He then proceeded to prove the accuracy of his deductions by demonstrating them upon the different planes of intelligence and substance ranging from the most abstract logical premise to the most concrete geometrical solid. From the common agreement of these diversified methods of proof he established the indisputable existence of certain natural laws.

(34) From thence floweth forth the Form of the Triad, being preexistent; not the first Essence, but that whereby all things are measured.

(1) After this we must narrate how, when he had admitted certain persons to be his disciples, he distributed them into different classes according to their respective merits. For it was not fit that all of them should equally participate of the same things, as they were naturally dissimilar; nor was it indeed right that some should participate of all the most honorable auditions, but others of none, or should not at all partake of them. For this would be uncommunicative and unjust. While therefore he imparted a convenient portion of his discourses to each, he benefited as much as possible all of them, and preserved the proportion of justice, by making each a partaker of the auditions according to his desert.

Hence, in conformity to this method, he called some of them Pythagoreans, but others Pythagorists; just as we denominate some men Attics, but others Atticists. Having therefore thus aptly divided their names, some of them he considered to be genuine, but he ordained that others should show themselves to be the emulators of these. He ordered therefore that with the Pythagoreans possessions should be shared in common, and that they should always live together; but that each of the others should possess his own property apart from the rest, and that assembling together in the same place, they should mutually be at leisure for the same pursuits. And thus each of these modes was derived from Pythagoras, and transmitted to his successors.

Again, there were also with the Pythagoreans two forms of philosophy; for there were likewise two genera of those that pursued it, the Acusmatici, and the Mathematici. Of these however the Mathematici are acknowledged to be Pythagoreans by the rest; but the Mathematici do not admit that the Acusmatici are so, or that they derived their instruction from Pythagoras, but from Hippasus. And with respect to Hippasus, some say that he was a Crotonian, but others a Metapontine. But the philosophy of the Acusmatici consists in auditions unaccompanied with demonstrations and a reasoning process; because it merely orders a thing to be done in a certain way, and that they should endeavour to preserve such other things as were said by him, as so many divine dogmas.

They however profess that they will not speak of them, and that they are not to be spoken of; but they conceive those of their sect to be the best furnished with wisdom, who retained what they had heard more than others. But all these auditions are divided into three species. For some of them indeed signify what a thing is; others what it especially is; but others, what ought, or what ought not, to be done. The auditions therefore which signify what a thing is, are such as, What are the islands of the blessed? The sun and moon. What is the oracle at Delphi? The tetractys. What is harmony? That in which the Syrens subsist . But the auditions which signify what a thing especially is, are such as, What is the most just thing?

To sacrifice. What is the wisest thing? Number. But the next to this in wisdom, is that which gives names to things. What is the wisest of the things that are with us, [i. e. which pertain to human concerns]? Medicine. What is the most beautiful? Harmony. What is the most powerful? Mental decision. What is the most excellent? Felicity. What is that which is most truly asserted? That men are depraved. Hence they say that Pythagoras praised the Salaminian poet Hippodomas, because he sings:

(6) It is likewise related of Clinias the Tarentine, that when he had learnt that Prorus the Cyrenæan, who was zealously addicted to the Pythagorean doctrines, was in danger of losing all his property, he sailed to Cyrene, after having collected a sum of money, and restored the affairs of Prorus to a better condition, not only incurring, in so doing, a diminution of his own property, but despising the peril which he was exposed to in the voyage. After the same manner also, Thestor Posidoniates, having learnt from report alone, that Thymaridas Parius the Pythagorean had fallen into poverty, from the possession of great wealth, is said to have sailed to Parus, after having collected a large sum of money, and thus reinstated Thymaridas in property.

These therefore are beautiful instances of friendship. The decisions, however, of the Pythagoreans respecting the communion of divine goods, the concord of intellect, and things pertaining to a divine soul, are much more admirable than the above examples. For they perpetually exhorted each other, not to divulse the God within them. Hence all the endeavour of their friendship both in deeds and words, was directed to a certain divine mixture, to a union with divinity, and to a communion with intellect and a divine soul. But it is not possible to find any thing better than this, either in what is uttered by words, or performed by deeds. For I am of opinion, that all the goods of friendship are comprehended in this. Hence, as we have collected in this, as in a summit, all the prerogatives of the Pythagoric friendship, we shall omit to say any thing further about it.

(7) Pythagoras likewise discovered another method of restraining men from injustice, through the judgment of souls, truly knowing indeed that this method may be taught, and also knowing that it is useful to the suppression of justice through fear. He asserted therefore, that it is much better to be injured than to kill a man; for that judgment is deposited in Hades, where the soul, and its essence, and the first nature of beings, are properly estimated. Being desirous, however, to exhibit in things unequal, without symmetry and infinite, a definite, equal, and commensurate justice, and to show how it ought to be exercised, he said, that justice resembles that figure, which is the only one among geometrical diagrams, that having indeed infinite compositions of figures, but dissimilarly disposed with reference to each other, yet has equal demonstrations of power.

Since also, there is a certain justice in making use of another person, such a mode of it as the following, is said to have been delivered by the Pythagoreans: Of associations with others, one kind is seasonable, but another is unseasonable. These likewise are distinguished from each other by difference of age, desert, the familiarity of alliance, and of beneficence, and whatever else there may be of the like kind in the different associations of men with each other. For there is a species of association, viz. of a younger with a younger person, which does not appear to be unseasonable; but that of a younger with an elderly person is unseasonable. For no species of anger, or threatening, or boldness, is becoming in a younger towards an elderly man, but all unseasonable conduct of this kind should be cautiously avoided.

A similar reasoning likewise should be adopted with respect to desert. For it is neither decorous, nor seasonable, to use an unrestrained freedom of speech, or to adopt any of the above-mentioned modes of conduct, towards a man who has arrived at the true dignity of consummate virtue. Conformably to this also, was what he said respecting the association with parents, and likewise with benefactors. He added, that there is a certain various and multiform use of an opportune time. For of those that are enraged and angry, some are so seasonably, but others unseasonably. And again, of those that aspire after, desire, and are impelled to any thing appetible, an opportune time is the attendant on some, and an unseasonable time on others.

And the same thing may be said concerning other passions and actions, dispositions, associations, and meetings. He farther observed, that an opportune time is to a certain extent , to be taught, and also, that what happens contrary to expectation, is capable of receiving an artificial discussion; but that when it is considered universally and simply, none of the above-mentioned particulars pertain to it. Nearly, however, such things are the attendants on it, as follow the nature of opportune time, viz. what is called the florid, the becoming, the adapted, and whatever else there may be homogeneous to these. He likewise asserted, that principle [or the beginning] is in the universe unity, and is the most honorable of things; and that in a similar manner it is so in science, in experience, and in generation.

And again, that the number two is most honorable in a house, in a city, in a camp, and in all such like systems. But that the nature of principle is difficult to be surveyed and apprehended in all the above-mentioned particulars. For in sciences, it is not the province of any casual understanding to learn and judge, by well surveying the parts of things, what the nature is of the principle of these. He added, that it makes a great difference, and that there is danger with respect to the knowledge of the whole of things, when principle is not rightly assumed. For none, in short, of the consequent conclusions can be sane, when the true principle is unknown.

The same thing may also be said respecting a principle of another kind. For neither can a house, or a city, be well instituted, unless each has a true ruler, who governs those that voluntarily submit to him. For it is necessary that in both these the governor should be willing to rule, and the governed to obey. Just as with respect to disciplines, when they are taught with proper effect, it is necessary that there should be a concurrence in the will both of the teacher and learner. For if there is a resistance on the part of either, the proposed work will never be accomplished in a proper manner. Thus therefore, he proved, that it was beautiful to be persuaded by rulers, and to be obedient to preceptors.

But he exhibited the following as the greatest argument through deeds, of the truth of his observations. He went from Italy to Delos, to Pherecydes the Syrian, who had been his preceptor, in order that he might afford him some assistance, as he was then afflicted with what is called the morbus pedicularis, and he carefully attended him to the time of his death, and piously performed whatever rites were due to his dead preceptor. So diligent was he in the discharge of his duties to him from whom he had received instruction.

(1) With respect to justice, however, we shall learn in the best manner, how he cultivated and delivered it to mankind, if we survey it from its first principle, and from what first causes it germinates, and also direct our attention to the first cause of injustice. For thus we shall discover how he avoided the latter, and what methods he adopted in order that the former might be properly ingenerated in the soul. The principle of justice therefore, is the common and the equal, through which, in a way most nearly approximating to one body and one soul, all men may be co-passive, and may call the same thing mine and thine; as is also testified by Plato, who learnt this from the Pythagoreans.

This therefore, Pythagoras effected in the best manner, exterminating every thing private in manners, but increasing that which is common as far as to ultimate possessions, which are the causes of sedition and tumult. For all things [with his disciples] were common and the same to all, and no one possessed any thing private. And he indeed, who approved of this communion, used common possessions in the most just manner; but he who did not, received his own property, which he brought to the common stock, with an addition to it, and departed. And thus he established justice in the best manner, from the first principle of it.

(6) Pythagoras thereupon discovered that the first and fourth strings when sounded together produced the harmonic interval of the octave, for doubling the weight had the same effect as halving the string. The tension of the first string being twice that of the fourth string, their ratio was said to be 2:1, or duple. By similar experimentation he ascertained that the first and third string produced the harmony of the diapente, or the interval of the fifth. The tension of the first string being half again as much as that of the third string, their ratio was said to be 3:2, or sesquialter. Likewise the second and fourth strings, having the same ratio as the first and third strings, yielded a diapente harmony. Continuing his investigation, Pythagoras discovered that the first and second strings produced the harmony of the diatessaron, or the interval of the third; and the tension of the first string being a third greater than that of the second string, their ratio was said to be 4:3, or sesquitercian. The third and fourth strings, having the same ratio as the first and second strings, produced another harmony of the diatessaron. According to Iamblichus, the second and third strings had the ratio of 8:9, or epogdoan.

(3) "The Pythagoreans indeed go farther than this, and honour even numbers and geometrical diagrams with the names and titles of the gods. Thus they call the equilateral triangle head-born Minerva and Tritogenia, because it may be equally divided by three perpendiculars drawn from each of the angles. So the unit they term Apollo, as to the number two they have affixed the name of strife and audaciousness, and to that of three, justice. For, as doing an injury is an extreme on the one side, and suffering one is an extreme on the on the one side, and suffering in the middle between them. In like manner the number thirty-six, their Tetractys, or sacred Quaternion, being composed of the first four odd numbers added to the first four even ones, as is commonly reported, is looked upon by them as the most solemn oath they can take, and called Kosmos." (Isis and Osiris.)

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

(19) And from the Antigone of Sophocles: "Bastardy is opprobrious in name; but the nature is equal;" And from the Aleuades of Sophocles: "Each good thing has its nature equal."

(1) Since, however, we are narrating the wisdom employed by Pythagoras in instructing his disciples, it will not be unappropriate to relate that which is proximate in a following order to this, viz. how he invented the harmonic science, and harmonic ratios. But for this purpose we must begin a little higher. Intently considering once, and reasoning with himself, whether it would be possible to devise a certain instrumental assistance to the hearing, which should be firm and unerring, such as the sight obtains through the compass and the rule, or, by Jupiter, through a dioptric instrument; or such as the touch obtains through the balance, or the contrivance of measures;—thus considering, as he was walking near a brazier’s shop, he heard from a certain divine casualty the hammers beating out a piece of iron on an anvil, and producing sounds that accorded with each other, one combination only excepted.

But he recognized in those sounds, the diapason, the diapente, and the diatessaron, harmony. He saw, however, that the sound which was between the diatessaron and the diapente was itself by itself dissonant, yet, nevertheless, gave completion to that which was the greater sound among them. Being delighted, therefore, to find that the thing which he was anxious to discover had succeeded to his wishes by divine assistance, he went into the brazier’s shop, and found by various experiments, that the difference of sound arose from the magnitude of the hammers, but not from the force of the strokes, nor from the figure of the hammers, nor from the transposition of the iron which was beaten. When, therefore, he had accurately examined the weights and the equal counterpoise of the hammers, he returned home, and fixed one stake diagonally to the walls, lest if there were many, a certain difference should arise from this circumstance, or in short, lest the peculiar nature of each of the stakes should cause a suspicion of mutation.

Afterwards, from this stake he suspended four chords consisting of the same materials, and of the same magnitude and thickness, and likewise equally twisted. To the extremity of each chord also he tied a weight. And when he had so contrived, that the chords were perfectly equal to each other in length, he afterwards alternately struck two chords at once, and found the before-mentioned symphonies, viz. a different symphony in a different combination. For he discovered that the chord which was stretched by the greatest weight, produced, when compared with that which was stretched by the smallest, the symphony diapason. But the former of these weights was twelve pounds, and the latter six. And, therefore, being in a duple ratio, it exhibited the consonance diapason; which the weights themselves rendered apparent.

But again, he found that the chord from which the greatest weight was suspended compared with that from which the weight next to the smallest depended, and which weight was eight pounds, produced the symphony diapente. Hence he discovered that this symphony is in a sesquialter ratio, in which ratio also the weights were to each other. And he found that the chord which was stretched by the greatest weight, produced, when compared with that which was next to it in weight, and was nine pounds, the symphony diatessaron, analogously to the weights. This ratio, therefore, he discovered to be sesquitertian; but that of the chord from which a weight of nine pounds was suspended, to the chord which had the smallest weight [or six pounds,] to be sesquialter.

For 9 is to 6 in a sesquialter ratio. In like manner, the chord next to that from which the smallest weight depended, was to that which had the smallest weight, in a sesquitertian ratio, [for it was the ratio of 8 to 6,] but to the chord which had the greatest weight, in a sesquialter ratio [for such is the ratio of 12 to 8.] Hence, that which is between the diapente and the diatessaron, and by which the diapente exceeds the diatessaron, is proved to be in an epogdoan ratio, or that of 9 to 8. But either way it may be proved that the diapason is a system consisting of the diapente in conjunction with the diatessaron, just as the duple ratio consists of the sesquialter and sesquitertian, as for instance, 12, 8, and 6; or conversely, of the diatessaron and the diapente, as in the duple ratio of the sesquitertian and sesquialter ratios, as for instance 12, 9, and 6.

After this manner, therefore, and in this order, having conformed both his hand and his hearing to the suspended weights, and having established according to them the ratio of the habitudes, he transferred by an easy artifice the common suspension of the chords from the diagonal stake to the limen of the instrument, which he called chordotonon . But he produced by the aid of pegs a tension of the chords analogous to that effected by the weights.

(4) There are three Things which the Mind has in it, and which rule it, yet the Mind in itself is the desirous Will. And those three Things, are three Kingdoms, or Principles; one is eternal, and the second is eternal, but the third is corruptible; the one has no Beginning; the second is without Beginning, eternally generated; and the third has a Beginning and End, and corrupts again [or perishes.]

(42) Iamblichus gathered thirty-nine of the symbolic sayings of Pythagoras and interpreted them. These have been translated from the Greek by Thomas Taylor. Aphorismic statement was one of the favorite methods of instruction used in the Pythagorean university of Crotona. Ten of the most representative of these aphorisms are reproduced below with a brief elucidation of their concealed meanings.

(24) Pythagoras taught that friendship was the truest and nearest perfect of all relationships. He declared that in Nature there was a friendship of all for all; of gods for men; of doctrines one for another; of the soul for the body; of the rational part for the irrational part; of philosophy for its theory; of men for one another; of countrymen for one another; that friendship also existed between strangers, between a man and his wife, his children, and his servants. All bonds without friendship were shackles, and there was no virtue in their maintenance. Pythagoras believed that relationships were essentially mental rather than physical, and that a stranger of sympathetic intellect was closer to him than a blood relation whose viewpoint was at variance with his own. Pythagoras defined knowledge as the fruitage of mental accumulation. He believed that it would be obtained in many ways, but principally through observation. Wisdom was the understanding of the source or cause of all things, and this could be secured only by raising the intellect to a point where it intuitively cognized the invisible manifesting outwardly through the visible, and thus became capable of bringing itself en rapport with the spirit of things rather than with their forms. The ultimate source that wisdom could cognize was the Monad, the mysterious permanent atom of the Pythagoreans.

(1) With respect to the amity, however, which subsists in all things towards all, Pythagoras delivered it in the clearest manner. And, the amity of the Gods indeed towards men, he unfolded through piety and scientific cultivation; but that of dogmas towards each other, and universally of the soul towards the body, and of the rational towards the species of the irrational part, through philosophy, and the theory pertaining to it. With respect to the amity of men also towards each other; that of citizens he delivered through sane legislation, but that of strangers through a correct physiology; and that between man and wife, or children, or brothers, and kindred, through unperverted communion. In short, he unfolded the friendship of all things towards all, and still farther, of certain irrational animals, through justice and a physical connexion and association.

But the pacification and conciliation of the body, which is of itself mortal, and of its latent contrary powers, he unfolded through health, and a diet and temperance conformable to this, in imitation of the salubrious condition of the mundane elements. In all these, however, Pythagoras is acknowledged to have been the inventor and legislator of the summary comprehension of them in one and the same name, which is that of friendship. And indeed he delivered such an admirable friendship to his associates, that even now those who are benevolent in the extreme towards each other, are said to belong to the Pythagoreans. It is necessary therefore to narrate the discipline of Pythagoras respecting these things, and the precepts which he used towards his disciples.