FRAGMENTS OF PHILOLAUS
From Boeckh
1. (Stob.21.7; Diog..#.8.85) The world's nature is a harmonious compound of infinite and finite elements; similar is the totality of the world in itself, and of all it contains. b. All beings are necessarily finite or infinite, or simultaneously finite and infinite; but they could not all be infinite only.
2. How, since it is clear that the beings can not be formed neither of elements that are all infinite, it is evident that the world in its totality, and its included beings are a harmonious compound of finite and infinite elements. That can be seen in works of art. Those that are composed of finite elements, are finite themselves, those that are composed of both finite and infinite elements, are both finite and infinite; and those composed of infinite elements, are infinite.
2. All things, at least those we know contain number; for it is evident that nothing whatever can either be thought or known, without number. 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.
3. The harmony is generally the result of contraries; for it is the unity of multiplicity, and the agreement of discordances. (Nicom.Arith.2:509)
4. This is the state of affairs about nature and harmony. The essence 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 are, and are known by us, should arrive to our knowledge, if this essence was not the internal foundation of the principles of which the world was founded, that is, of the finite and infinite elements. How since these principles are not mutually similar, neither of similar nature, it would be impossible that the order of the world should have been formed by them, unless the harmony intervened, in any manner whatever. Of course, the things that were similar and of similar nature, did not need the 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 is a fourth, plus a fifth. The fifth is greater than the fourth by nine eighths; for from the lowest string to the second lowest, there is a fourth; and from this to the next, a fifth; but from this to the next, or "third," a fourth; and from this "third" to the lowest, a fifth. The interval between the second lowest and the "third" (from the top) is nine eighths; the interval of the fourth, is four thirds; that of the fifth, three halves; that of the octave, the double relation. Thus the harmony contains five nine-eighths plus two sharps; the fifth, three nine eighths, plus one sharp; the fourth two nine-eighths, plus one.
6. (Boethius, Music, 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 are 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 eighth 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 period, 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 semi-tone, 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 coma, and that 27 unities form an entire tone, because 27 is the difference between 215 and 243, which are distant by one tone.
7. (Boethius, Music, 3:8). These are the definitions that Philolaus has given of these intervals, and of still smaller intervals. The coma, says he, is the interval whose eighth-ninths relation exceeds the sum of two sharps, namely, the sum of two minor semi-tones. The schisma is half the comma, the diaschisma is half the sharp, namely, of the minor semi-tone.
8. (Claudius Namert.de Stat. anim.2:3) Before treating of the substance of the soul, Philolaus, according to geometrical principles, treats of music, arithmetic, measures, weights, numbers, insisting that these are the principles which support the existence of the Universe.
9. (Nicom.. Arith.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 has always 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 square, has been named, by the Pythagorean Philolaus the name of geometrical harmony, because he thinks he recognizes in it all the hamonic relations.
10. (Stob. Eclogl.1:5:7:p.360) The world is single; it began to form from the centre outwards. Starting from this centre, the top is entirely identical to the base; still you might say that what is above the centre is opposed to what is below it; for, for the base, lowest point would be the centre, as for the top, the highest point would still be the centre; and likewise for the other parts; in fact, in respect to the centre, each one of the opposite points is identical, unless the whole be moved. b.(Stob.Ecl.l:2l:3:p.468) The prime composite, the One placed in the centre of the sphere is called Hestia.
11. a. (Stob.Ecl.l:22:l:p.488) Philolaus has located the fire in the middle, the centre; he calls it Hestia, of the All, the house [policeipest] of Jupiter, and the mother of the Gods, the altar, the link, the measure of nature. Besides, he locates a second fire, quite at the top, surrounding the world. The centre, says he, is by its nature the first; around it, the ten different bodies carry out their choric dance; these are, the heaven, the planets, lower the sun, and below it the moon ; lower the earth, and beneath this, the anti-earth (a body invented by the Pythagoreans, says Aristotle, Met i: 5) then beneath these bodies the fire of Hestia, in the centre, 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 revolutionary circle of Olympus, and where in order are disposed the five planets, the sun and moon, forms the Cosmos world; finally, beneath the latter is the sublunar region, which surrounds the earth, where are the generative things susceptible to change; 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. (Plut. Plac.Phil.3:ll). The Pythagorean Philolaus located the fire in the centre, it is the Hestia of the All, then the Anti-earth, then the earth we inhabit, placed opposite the other, and moving circularly; which is the cause that its inhabitants are not visible to ours. c. (Stob.Ecl.l:21:6:p.452). The directing fire, [of] 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. (Plut.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 noon, which is driven away therefrom by the circulation of air; the loss of these two stars nourish the world.
13. (Diog.Laert.8:85). Philolaus was the first who said the world moves in a circle; others attribute it to Hivatas of Syracuse. b. (Plut.Plac.Philos.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. (Stob.Ecl.l: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, after 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 a mirror. Perhaps we might consider as a third light that which, from the mirror in which it reflects, and falls back on us in dispersed rays.
15. (Stob.Eclog.1:26: l:p. 562) Some Pythagoreans, among whom is Philolaus, pretend 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 pretend that the apparent form of the noon is only the reflection of the sea, which we inhabit, which passes beyond the circle of fire.
16. (Censorinus, de Die Natal.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. (Iambl.ad Nicon.Arith.11). Philolaus says that number is the sovereign and autogenic force which maintains the eternal permanence of cosmic things.
18. (Stob.1:3:8). The power, efficacy and essence of number is seen in the decad; it is great, it realizes all its purposes, 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 infinite, obscure, and furtive. Indeed it is the nature of number which teaches us comprehension, which serves us as guide, which teaches us all things, which would remain impenetrable and unknown for every man, for there is nobody who could get so clear a notion about it, things in themselves, neither 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, inidividualises them, both in finite and infinite things. And it is not only in matters pertaining to genii 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 production of arts and music. The nature of number and harmony are numberless, for what is false has no part in their [-------]; for the principle of error and envy is thoughtless, irrational, infinite nature. Never could error slip into number; for its nature is hostile thereto. Truth is the proper, innate character of number.
b. (Theologoumena, 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, Platon.Nemn.p.49) The [Tetractys] 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. (Lucien, Pro. Laps. Inter. Salut. 5.) Some called the Tetractys 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. (Theon of Smyrna, Plat. Math. 4.). Archytas and Philolaus use the terms monad and unity interchangeably.
b. (Syrianus, sub init, Comment. in Arist. Net. I.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 finite and the infinite. He shows that it is by the limit, that every coordinate series of things further approaches Unity, and that it is by infinity 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 clause 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. (Iambl. ad Nicom. Arith. p109). In the formation of square numbers by addition, unity is as it were the starting-post from which one starts, 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, and the root is like the turning-point where the horses turn to go back through similar numbers to unity, as in the square of 5.
For example:
1-----2-----3-----4
-----5 = added, 25
1-----2-----3-----4It is not the same with rectangular numbers; if, just as if 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. If you 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 Philolaus (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 1------2------3-----4 ------------------ 5= 24 4 --------2-----3-----4 -------------------
d. (Ohilo, Mundi 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, different from other things.
e. (Athenag.Legat.pro Christo). Philolaus says that all things are by God kept as if in captivity, and thereby implies, that He is single and superior to matter.
20. (Proclus, ad Euclid. Elem.I.33). Even among the Pythagoreans we find different angles consecrated to the different divinities, as did Philolaus, who devoted 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 are [have] angles which unite the lines, and are their [ends]. Philolaus was therefore right in devoting the angle of the triangle to four divinities, [Cronos], Hades, Mars and Bacchus, under these 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, Cronos presides over everything humid and cold essence; Mars, over everything fiery; Hades contains everything terrestrial, and Dionysius directs the generation of wet and warm things, symboled 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 tetragone 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 organisation of the being that is solid in its totality, and of its pure and immovable essence. They were therefore right to express it symbolically by the figure the tetragon. Besides, Philolaus, with another stroke of genius, calls the angle of the tetragon. that of Rhea, of Dimeter, and of Hestia...For considering the earth as a tetragon, and noting that this element possesses the property of continuousness, as we learned it from Timaeus, and 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 causes. 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; 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 beings. Their product is the dodecad, which ends in the single monad, the sovereign principle of Jupiter; for Philolaus says that the angle of the dodecagon belongs to Jupiter, because in unity Jupiter contains the entire of the dodecad.
21. (Theolog.Arithm. 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 reflexion are communicated to beings by the number eight.
b. (Theolog.Arithm. p.22). There are four principles of the reasonable animal, as Philolaus 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 facility of striking roots and reproducing the first being; the sexual organs, of the faculty of projecting the sperm, and procreating. The skull contains the principle of man, the heart of the animal, the navel that of the plant, the sexual organs that of all living beings, for these grow and produce offspring.
c. (Stob.Edog.Physic.l:2:3: p.l0). There are bodies five in the sphere; fire, water, earth, air; and the circle of the sphere, which makes the fifth.
22. (Stob.Ecog.l:2:2: p.418). 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 single, 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 to the moon, 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 reason and the 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 save their essence and form, thanks to generation, which reproduces the identical form of the father who has begotten and fashioned them.
23. (Claudian Mamort. De Statu. Anim.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.
b. (Macrob. Dream of Scipio, I:04). Plato says that the soul is a self-moving essence; Xenocrates defines the soul as a self-moving number; Aristotle calls it an entelechy; and Pythagoras and Philolaus, a harmony.
c. (Olympiod. ad 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. (Clem.Strom. 3: p.433). It will help us to remember the Pythagorean Philolaus's 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.
24. (Arist.Eth.Eud. 2:8). As Philolaus has said, there are some reasons stronger than us. b. (Iambl. ad Nicom. Arithm. 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. (Sext. Empir, Adv.Math. 7:92: p. 388). Anaxagoras has said how reason in general is the faculty 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 insist that if this Reason is capable of undestanding All, it is only that its essence is kindred with this nature, for it is in the nature of things that the similar be understood by the similar.
26. (Laurent.Lydus,de Mens.p 16; Cedrenus --; 169b). Philolaus was therefore right in calling it a decad, because it receives (pun) the Infinite, and Prpheus was right in calling it the branch, because it is the branch from which issue all the numbers, as do many branches.
b. (Cedrenus, l.p72). Philolaus was therefore right to say that the number seven was motherless.
c. (Cedrenus, l.p.23). Philolaus was therefore right to call the spouse of Kronos, the Dyad.