THE PYTHAGOREAN SOURCEBOOK AND LIBRARY
FIGURE 13. THE MUSIC OF THE SPHERES
THE FRAGMENTS OF PHILOLAUS
PHILOLAUS OF TARENTUM (latter half of fifth century B.C.E.) was educated by Lysis, one of the two Pythagoreans who escaped the persecution of the school at Croton. He was the first member of the school to record Pythagorean teachings in writing and it appears that these writings influenced the thought of Plato. Moreover, Plato's nephew Speusippus who took over the leadership of the Academy, drew on the work of Philolaus in compiling his treatise On Pythagorean Numbers, which dealt primarily with the properties of the Decad.
The fragments of Philolaus assembled here deal with the Pythagorean theory of the Limited and Unlimited, the principle of Harmonia through which they are conjoined, and the primacy of Number in the nature of Harmonia, and also the importance of Number in the pursuit of knowledge.
The Philolaic fragments are generally accepted as being authentic and their significance is not to be underestimated. This is because of all surviving Pythagorean writings the fragments of Philolaus are the earliest and most faithfully reflect the teachings of the original Pythagorean school.
For a good discussion of Philolaus see section 44 in Kathleen Freeman's Companion to the Presocratic Philosophers. The numbers following the fragments (e.g., DK 1) refer to the numeration in Diels-Kranz, Fragmente der Vorsokratieker, and are translated in Kathleen Freeman's Ancilla to the Pre-Socratic Philosophers.
THE LIFE OF PHILOLAUS
FROM DIOGENES LAERTIUS
PHILOLAUS OF CROTON, a Pythagorean, was he from whom Plato, in some of his Letters, begged Dion to purchase Pythagorean books. He (Dion) died under the accusation of having had designs on the tyranny. I have made about him the following epigram: "I advise everybody to take good care to avoid suspicion; even if you are not guilty but seem so, you are ruined. That is why Croton, the homeland of Philolaus, destroyed him, because he was suspected of wishing to establish autocracy."
Philolaus teaches that all things are produced by necessity and harmony, and he is the first who said that the earth has a circular movement; others, however, insist this was due to Hicetas of Syracuse. He had written a single book which the philosopher Plato, visiting Dionysius in Sicily, bought, according to Hermippus, from Philolaus' parents, for the sum of 40 Alexandrian minae, whence he drew his Timaeus. Others state that he received them as a present for having obtained the liberty of one Philolaus' disciples, whom Dionysius had imprisoned. In his Homonyms, Demetrius claims that he is the first of the Pythagorean philosophers who made a work On Nature public property. This book begins as follows: "The world's being is the harmonious compound of Unlimited and Limiting principles; such is the totality of the world and all it contains."
THE FRAGMENTS OF PHILOLAUS
1. (Stobaeus, 21. 7; Diogenes Laertius, 8. 85) The world's nature is a harmonious compound of Limited and Unlimited elements; similar is the totality of the world in itself, and of all it contains (DK 1).
B. All beings are necessarily Limited or Unlimited, or simultaneously Limited and Unlimited; but they could not all be Unlimited only.
2. Now, since it is clear that the beings cannot be formed either of elements that are all Unlimited, it is evident that the world in its totality, and its included beings are a harmonious compound of Limited and Unlimited elements. That can be seen in existing things. Those that are composed of Limiting elements, are Limited themselves; those that are composed of both Limiting and Unlimited elements, are both Limited and Unlimited; and those composed of Unlimited elements are Unlimited (DK 2).
B. All things, at least those we know, contain Number; for it is evident that nothing whatever can either be thought or known, without Number (DK 4). Number has two distinct kinds: the odd, and the even, and a third, derived from a mingling of the other two kinds, the even-odd. Each of its subspecies is susceptible of many very numerous varieties, which each manifests individually (DK 5).
3. (Nicomachus, Arith. Intr., 2. 509) Harmony is generally the result of contraries; for it is the unity of multiplicity, and the agreement of discordances (DK 10).
4. This is the state of affairs concerning Nature and Harmony. The Being of things is eternal; it is a unique and divine nature, the knowledge of which does not belong to man. Still it would not be possible that any of the things that exist, and that are known by us, should arrive to our knowledge if this Being was not the internal foundation of principles of which the world was founded-that is, of the Limited and Unlimited elements. Now since these principles are not mutually similar, nor of similar nature, it would be impossible that the order of the world should have been formed by them in any manner whatever unless harmony had intervened. Of course, the things that were similar, and of similar nature, did not need harmony; but the dissimilar things, which have neither a similar nature, nor an equivalent function, must be organized by the harmony, if they are to take their place in the connected totality of the world.
5. The extent of the Harmony [octave] is a fourth, plus a fifth. The fifth is greater than the fourth by 8:9; for from the lowest string to the second lowest there is a fourth; and from this to the higher a fifth; but from this to the next, or third string, a fourth; and from this third string to the lowest, a fifth. The interval between the second lowest and the third [from the bottom] is 8:9 [a tone]; the interval of the fourth is 3:4; that of the fifth, 2:3; that of the octave, 1:2. Thus the Harmony contains five whole tones plus two semitones; the fifth, three tones, plus one semitone; the fourth, two whole tones, plus one semitone (DK 6).
6. (Boethius, De. Inst. Mus., 3. 5). Nevertheless the Pythagorean Philolaus has tried to divide the tone otherwise; his tone's starting-point is the first uneven number which forms a cube, and you know that the first uneven number was an object of veneration among these Pythagoreans. Now the first odd number is three; thrice three is nine, and nine times three is 27, which differs from the number 24 by the interval of one tone, and differs from it by this very number 3. Indeed, 3 is one eight of 24, and this eighth part of 24, added to 24 itself, produces 27, the cube of 3. Philolaus divides this number 27 in two parts, the one greater than half, which he calls apotome the other one smaller than half he calls sharp, but which latterly has become known as minor half-tone. He supposes that this sharp contains thirteen unities, because 13 is the difference between 256 and 243, and that this same number is the sum of 9, 3, and unity, in which the unity plays the part of the point, 3 of the first odd line, and 9 of the first odd square. After having, for these reasons, expressed by 13 the sharp, which is called a semitone, out of 14 unities he forms the other part of the number 27, which he calls apotome, and as the difference between 13 and 14 is the unity, he insists that the unity forms the comma, and that 27 unities form an entire tone, because 27 is the difference between 216 and 243, which are distant by one tone. 
7. (Boethius, De. Inst. Mus., 3. 8). These are the definitions that Philolaus has given of these intervals, and of still smaller intervals. The comma, says he, is the interval whose eight-ninths relation exceeds the sum of two sharps, namely, the sum of two minor semitones. The schisma is half the comma, the diaschisma is half the sharp, namely, of the minor semitone.
8. (Claudianus Mamertus, De Statu Animae, 2. 3). Before treating of the substance of the soul, Philolaus, according to geometrical principles, treats of music, arithmetic, measures, weights, and numbers, insisting that these are the principles which support the existence of the universe.
9. (Nicomachus, Arithm. Intr., 2. p. 72). Some, in this following Philolaus, think that this kind of a proportion is called harmonic, because it has the greatest analogy with what is called geometrical harmony; which is the cube, because all its dimensions are mutually equal, and consequently in perfect harmony. Indeed this proportion is revealed in all kinds of cubes which always have 12 sides, 8 angles, and 6 surfaces. 
B. (Cassiodorus, Exp. in Ps., 9, p. 36). The number 8, which the arithmeticians call the first actual cube, has been given by the Pythagorean Philolaus the name of geometrical harmony, because he thinks he recognizes in it all the harmonic relations.
10 A. (Stobaeus, Eclog. Physic., 1. 15.7. p. 360). The world is single and it came into being from the center outwards. Starting from this center, the top is entirely identical to the base; still you might say that what is above the center is opposed to what is below it; for the base, lowest point would be the center, as for the top, the highest point would still be the center; and likewise for the other parts; in fact, in respect to the center, each one of the opposite points is identical, unless the whole be moved (DK 17).
B. (Stobaeus, Eclog. Physic., 1. 21. 1. p. 468). The prime composite, the One placed in the center of the sphere, is called Hestia (DK 7).
11. A. (Stobaeus, Eclog. Physic., 1. 22. 1. p. 488). Philolaus has located the fire in the middle, the center; he calls it Hestia, of the All, the Guardpost of Zeus, the Mother of the Gods, the Altar, the Link, and the Measure of Nature. Besides, he locates a second fire, quite at the top, surrounding the world. The center, says he, is by its nature the first; around it, the ten different bodies carry out their choral dance. These are: the heaven, the planets, lower the sun, and below it the moon; lower the earth, and beneath this, the counter-earth, then beneath these bodies the fire of Hestia, in the center, where it maintains order. The highest part of the Covering, in which he asserts that the elements exist in a perfectly pure condition, is called Olympus; the space beneath the revolution-circle of Olympus, and where in order are disposed the five planets, the sun and moon, forms the Cosmos; finally, beneath the latter is the sublunar region, which surrounds the earth, where are the generative things, susceptible to change. All that is the heaven. The order which manifests in the celestial phenomena is the object of science; the disorder which manifests in the things of becoming, is the object of virtue; the former is perfect, the latter is imperfect.
B. (ps.-Plutarch, Plac. Phil., 3. 11). The Pythagorean Philolaus locates the fire in the center -- it is the Hearth (Hestia) of the All -- then the counter-earth, then the earth we inhabit, placed opposite the other, and moving circularly, which is the reason that its inhabitants are not visible to ours.
C. (Stobaeus, Eclog. Physic., 1. 21. 6. p. 452). The directing fire, says Philolaus, is in the entirely central fire, which the Demiurge has placed as a sort of keel to serve as foundation to the sphere of the All.
12. (ps.-Plutarch, Plac. Phil., 2. 5). Philolaus explains destruction by two causes: one is the fire which descends from heaven, the other is the water of the moon, which is driven away therefrom by the circulation of the air; the exhalations of these two stars nourish the world.
13. A. (Diogenes Laertius, 8.85). Philolaus was the first who said that the world moves in a circle; others attribute it to Hicetas of Syracuse.
B. (ps.-Plutarch, Plac. Phil., 3. 7). Some insist that the earth is immovable; but the Pythagorean Philolaus says that it moves circularly around the central fire, in an oblique circle like the sun and moon.
14. (Stobaeus, Eclog. Physic., 1. 25. 3. p. 530). The Pythagorean Philolaus says that the sun is a vitrescent body which receives the light reflected by the fire of the Cosmos, and sends it back to us, having filtered them, light and heat; so that you might say that there are two suns, the body of the fire which is in the heaven, and the igneous light which emanates therefrom, and reflects itself in a kind of mirror. Perhaps we might consider as a third light that which, from the mirror in which it is reflected, falls back on us in dispersed rays.
15. (Stobaeus, Eclog. Physic., 1. 26. 1. p. 562). Some Pythagoreans, among whom is Philolaus, suggest that the moon's resemblance to the earth consists in its surface being inhabited, like our earth, but by animals and vegetation larger and more beautiful. For the lunar animals are fifteen times larger than ours, and do not evacuate excreta. The day is also fifteen times as long. Others say that the apparent form of the moon is only the reflection of the sea, which we inhabit and which passes beyond the circle of fire.
16. (Censorinus, De Die Natali, 18). According to the Pythagorean Philolaus there is a year composed of 59 years and 21 intercalary months; he considers that the natural year has 364 and a half days.
17. (Iamblichus, In. Nicom., 11). Philolaus says that Number is the sovereign and autogenic force which maintains the eternal permanence of cosmic things (DK 23).
18. A. (Stobaeus, 1. 3. 8). The power, efficacy and essence of Number is seen in the Decad; it is great, it realizes all its purposes, and it is the cause of all effects. The power of the Decad is the principle and guide of all life, divine, celestial, or human into which it is insinuated; without it everything is unlimited, obscure, and furtive. Indeed, it is the nature of Number which teaches us comprehension, which serves us as guide, and teaches us all things which would otherwise remain impenetrable and unknown to every man. For there is nobody who could get a clear notion about things in themselves, nor in their relations, if there was no Number or Number-essence. By means of sensation. Number instills a certain proportion. and thereby establishes among all things harmonic relations, analogous to the nature of the geometric figure called the gnomon; it incorporates intelligible reasons of things, separates them, individualizes them. both in limited and unlimited things. And it is not only in matters pertaining to daimons or Gods that you may see the force manifested by the nature and power of Number, but it is in all its works, in all human thoughts, everywhere indeed, and even in the productions of arts and music. The nature of Number and Harmony are numberless, for what is false has no part in their essence and the principle of error and envy is thoughtless, irrational, indefinite nature. Never could error slip into Number, for its nature is hostile thereto. Truth is the proper, innate character of Number (DK 11).
B. (ps.-Iamblichus, Theologumena Arithmeticae, 61). The Decad is also named Faith, because, according to Philolaus, it is by the Decad and its elements, if utilized energetically and without negligence, that we arrive at a solidly grounded faith about beings. It is also the source of memory, and that is why the Monad has been called Mnemosyne.
C. (Theon of Smyrna, Plat. Math., p. 49). The Decad determines every number, including the nature of everything, of the even and the odd, of the mobile and immobile, of good and evil. It has been the subject of long discussions by Archytas, and of Philolaus in his work On Nature.
D. (Lucian, Pro Lapsu Inter Salutandum, 5). Some called the Tetraktys the great oath of the Pythagoreans, because they considered it the perfect number, or even because it is the principle of Health; among them is Philolaus.
19. A. (Theon of Smyrna, Plat. Math., 4). Archytas and Philolaus use the terms Monad and Unity interchangeably.
B. (Syrianus, sub. init., Comment. in Arist. Met., 1. xiv). You must not suppose that the philosophers begin by principles supposed to be opposite; they know the principle above these two elements, as Philolaus acknowledges when saying that it is God who hypostasizes the Limited and Unlimited. He shows that it is by Limit that every coordinate series of things further approaches Unity, and that it is by the Unlimited that the lower series is produced. Thus even above these two principles they posited the unique and separate cause distinguished by all of its excellence. This is the cause which Archinetus called the cause before the cause, and which Philolaus vehemently insists is the principle of all, and of which Brontinus says that in power and dignity it surpasses all reason and essence.
C. (Iamblichus, In. Nicom., p. 109). In the formation of square numbers by addition, unity is as it were the starting-post from which one begins, and also the end whither one returns; for if one places the numbers in the form of a double procession, and you see them grow from unity to the root of the square, then the root is like the turning-point where the horses tum to go back through similar numbers to unity, as in the square of 5. For example:
It is not the same with rectangular numbers. If, just as in the gnomon, one adds to any number the sum of the even, then the number two will alone seem to receive and stand addition, and without the number two it will not be possible to produce rectangular numbers. Ifyou set out the naturally increasing series of numbers in the order of the double race-track, then unity, being the principle of everything, according to Philo1aus (for it is he who said, "unity, the principle of everything"), will indeed present itself as the barrier, the starting point which produces the rectangular numbers, but it will not be the goal or limit where the series returns and comes back; it is not unity, but the number 2, which will fulfill this function. Thus, 6 x 4:
D. (Philo, Mund. Opif., 24). Philolaus confirms what I have just said by the following words; "He who commands and governs everything is a God who is single, eternally existing, immutable, self-identical, and different from other things."
E. (Athenagoras, Legat. Pro Christ., 6). Philolaus says that all things are by God kept as in captivity, and thereby implies the he is single and superior to matter to matter.
20. (Proclus, In Eucl., I. 36). Even among the Pythagoreans we find different angles consecrated to the different divinities, as did Philolaus, who attributed to some the angle of the triangle, to others the angle of the rectangle, to others other angles, and sometimes the same to several.
The Pythagoreans say that the triangle is the absolute principle of generation of begotten things, and of their form; that is why Timaeus says that the reasons of physical being, and of the regular formation of the elements are triangular; indeed, they have the three dimensions, in unity they gather the elements which in themselves are absolutely divided and changing; they are filled with the infinity characteristic of matter, and above the material beings they form bonds that indeed are frail. That is why triangles are bounded by straight lines, and have angles which unite the lines, and are their bonds. Philo1aus was therefore right in devoting the angle of the triangle to four divinities, Kronos, Hades, Mars, and Bacchus, under these four names combining the fourfold disposition of the elements, which refers to the superior part of the universe, starting from the sky, or sections of the zodiac. Indeed, Kronos presides over everything humid and cold in essence; Mars, over everything fiery; Hades contains everything terrestrial, and Dionysius directs the generation of wet and warm things, represented by wine, which is liquid and warm. These four divinities divide their secondary operations, but they remain united; that is why Philolaus, by attributing to them one angle only, wished to express this power of unification.
The Pythagoreans also claim that, in preference to the quadrilateral, the tetragon bears the divine impress, and by it they express perfect order. For the property of being straight imitates the power of immutability, and equality represents that of permanence; for motion is the result of inequality, and rest, that of equality. Those are the causes of the organization of the being that is solid in its totality, and of its pure and immovable essences. They were therefore right to express it symbolically by the figure of the tetragon. Besides, Philolaus, with another stroke of genius, calls the angle of the tetragon that of Rhea, of Demeter, and of Hestia.... For considering the earth as a tetragon, and noting that this element possesses the property of continuousness, as we learned from Timaeus, and that the earth receives all that drips from the divinities, and also the generative powers that they contain, he was right in consecrating the angle of the tetragon to these divinities which procreate life. Indeed, some of them call the earth Hestia and Demeter, and claim that it partakes of Rhea, in its entirety, and that Rhea contains all the begotten cause. That is why, in obscure language, he says that the angle of the tetragon contains the single power which produces the unity of these divine creations.
And we must not forget that Philolaus assigns the angle of the triangle to four divinities, and the angle of the tetragon to three, thereby indicating their penetrative faculty, whereby they influence each other mutually, and showing how all things participate in all things, the odd things in the even and the even in the odd. The triad and the tetrad, participating in the generative and creative beings, contain the whole regular organization of begotten things. Their product is the dodecad, which ends in the single monad, the sovereign principle of Zeus, for Philolaus says that the angle of the dodecagon belongs to Zeus, because in unity Zeus contains the entire number of the dodecad.
21 A. (ps.-Iamblichus, Theologumena Arithmeticae, p. 56). After the mathematical magnitude which by its three dimensions or intervals realizes the number Four, Philolaus shows us the being manifesting in number Five quality and color; in the number Six, the soul and life; in the number Seven, reason, health, and what he calls light; then he adds that love, friendship, prudence and reflection are communicated to beings by the number Eight.
B. (ps.-Iamblichus, Theologumena Arithmeticae, p. 22). There are four principles of the reasonable animal, as Philolaus says in his work On Nature: the skull, the heart, the navel, and the sexual organs. The head is the seat of reason; the heart, that of the soul or life, and sensation; the navel, the principle of the faculty of striking roots and reproducing the first being; the sexual organs, of the faculty of projecting the sperma, and procreating. The skull contains the principle of man, the heart that of the animal, the navel that of the plant, the sexual organs that of all living beings, for these grow and produce offspring (DK 13).
C. (Stobaeus, Eclog. Physic., 1. 2. 3. p. 10). There are five bodies in the sphere: fire, water, earth, air, and the circle of the sphere which makes the fifth (DK 12). 
22. (Stobaeus, Eclog. Physic., 1. 20. 2. p. 418). This is from the Pythagorean Philolaus, drawn from his book On the Soul. He insists that the world is indestructible. Here is what he says in his book On the Soul:
That is why the world remains eternally, because it cannot be destroyed by any other, nor spontaneously destroy itself. Neither within it, nor without it can be found a force greater than itself, able to destroy it. The world has existed from all eternity, and will remain eternally, because it is One, governed by a principle whose nature is similar to its own, and whose force is omnipotent and sovereign. Besides, the single world is continuous, and endowed with a natural respiration, moving eternally in a circle, having the principle of motion and change; one of its parts is immovable, the other is changing. The immovable part extends from the soul, that embraces everything, to the moon; and the changing part from the moon to the earth; or, since the Mover has been acting since eternity, and continues his action eternally, and since the changeable part receives its manner of being from the Mover who acts thereon, it necessarily results thence that one of the parts of the world ever impresses motion, and that the other ever receives it passively. The one is entirely the domain of Mind and Soul, the other of Generation and Change; the one is anterior in power, and superior, the other is posterior and subordinate. The composite of these two things, the divine eternally in motion, and of generation ever changing, is the World. That is why one is right in saying that the world is the eternal energy of God, and of becoming which obeys the laws of changing nature. The one remains eternally in the same state, self-identical; the remainder constitutes the domain of plurality, which is born and perishes. But nevertheless, the things that perish transmit their essence and form, thanks to generation, which reproduces the identical form of the father who has begotten and fashioned them (DK 21).
23 A. (Claudianus Mamertus, De Statu Animae, 2. p. 7). The soul is introduced and associated with the body by Number, and by a harmony simultaneously immortal and incorporeal....the soul cherishes its body, because without it the soul cannot feel; but when death has separated the soul therefrom, the soul lives an incorporeal existence in the cosmos (DK 22).
B. (Macrobius, Commentariorum in Somnium Scipionis, 1. 14). Plato says that the soul is a self-moving essence; Xenocrates defines the soul as a self-moving number; Aristotle called it an entelechy; and Pythagoras and Philolaus, a harmony.
C. (Olympiodorus, In. Plat. Phaed., p. 150). Philolaus opposed suicide because it was a Pythagorean precept not to lay down the burden, but to help others carry theirs; namely, that you must assist, and not hinder it.
D. (Clement of Alexandria, Stromateis, 3. p. 433). It will help us to remember the Pythagorean Philolaus' utterance that the ancient theologians and divines claimed that the soul is bound to the body as a punishment, and is buried in it as in a tomb (DK 14).
24. A. (Aristotle, Eth. Eud., 2. 9). As Philolaus has said, there are some reasons (logoi) stronger than us (DK 16).
B. (Iamblichus, In Nicom., 1. 25). I shall later have a better opportunity to consider how, in raising a number to its square, by the position of the simple component unities, we arrive at very evident propositions, naturally, and not by any law, as says Philolaus.
25. (Sextus Empiricus, Adversus Mathematicos, 7. 92. p. 388). Anaxagoras has said how reason in general is the faculty of discerning and judging; the Pythagoreans also agree that it is Reason, not reason in general, but the Reason that develops in men by the study of mathematics, as Philolaus used to say, and they insist that if this Reason is capable of understanding All, it is only because its essence is kindred with this nature, for it is in the nature of things that the similar be understood by the similar.
26 A. (Laurentius Lydus, De Mens., p. 16; Cedrenus, 1. 169b). Philolaus was therefore right in calling it a Decad, because it receives (dechomai) the Infinite, and Orpheus was right in calling it the Branch, because it is the branch from which issue all the numbers, as so many branches.
B. (Cedrenus, 1. p. 72). Philolaus was therefore right to say that the number seven was motherless (DK 20).
C. (Cedrenus, 1. p. 208). Philolaus was therefore right to call the Dyad the spouse of Kronos (DK 20A).
1. This fragment concerns itself with musical intervals smaller than a whole tone. For a good discussion, which also relates to the next fragment, see McClain, The Pythagorean Plato. 159-62.
2. 8 is the harmonic mean between 6 and 12.
3. There is reason to believe that the "five bodies" referred to in this fragment are the so-called regular polyhedra or "Platonic solids" which are described for the first time in Plato's Timaeus. It is quite likely that the earlier Pythagoreans were aware of the regular solids. If this fragment is genuine, Plato followed their lead in associating these "molecular" forms with the "elements" of Greek physics, the latter of which actually represent slates of matter rather than specific substances.