Nature in the universe was fitted together from the Non-Limited and the Limiting, both the universe as a whole and everything in it.

All existing things must necessarily be either Limiting, or Non-Limited, or both Limiting and Non-Limited. But they could not be merely Non-Limited (nor merely Limited). Since however it is plain that they are neither wholly from the Limiting nor wholly from the Non-Limited, clearly then the universe and its contents were fitted together from both the Limiting and the Non-Limited. This is proved also by actual existing things; for those of them which are made of Limiting (elements) impose Limit, whereas those made of both Limiting and Non-Limited (elements) both do and do not impose Limit, and those made of Non-Limited (elements) will appear Non-Limited.

For there could not even be an object set before knowledge to begin with, if all things were Non-Limited.

Actually, everything that can be known has a Number; for it is impossible to grasp anything with the mind or to recognise it without this (Number).

Actually, Number has two distinct forms, odd and even, and a third compounded of both, the even-odd; each of these two forms has many aspects, which each separate object demonstrates in itself.

This is how it is with Nature and Harmony: the Being of things is eternal, and Nature itself requires divine and not human intelligence; moreover, it would be impossible for any existing thing to be even recognised by us if there did not exist the basic Being of the things from which the universe was composed, (namely) both the Limiting and the Non-Limited. But since these Elements exist as unlike and unrelated, it would clearly be impossible for a universe to be created with them unless a harmony was added, in which way this (harmony) did come into being. Now the things which were like and related needed no harmony; but the things which were unlike and unrelated and unequally arranged are necessarily fastened together by such a harmony, through which they are destined to endure in the universe.

The content of the Harmony (Octave) is the major fourth and the major fifth; the fifth is greater than the fourth by a whole tone; for from the highest string (lowest note) to the middle is a fourth, and from the middle to the lowest string (highest note) is a fifth. From the lowest to the third string is a fourth, from the third to the highest string is a fifth. Between the middle and third strings is a tone. The major fourth has the ratio 3:4, the fifth 2:3, and the octave 1:2. Thus the Harmony (Octave) consists of five whole tones and two semitones, the fifth consists of three tones and a semitone, and the fourth consists of two tones and a semitone.

The first composite (entity), the One, which is in the centre of the Sphere, is called Hearth.

The One is the beginning of everything.

By nature, not by convention.

Harmony is a Unity of many mixed (elements), and an agreement between disagreeing (elements).

One must study the activities and the essence of Number in accordance with the power existing in the Decad (Ten-ness); for it (the Decad) is great, complete, all-achieving, and the origin of divine and human life and its Leader; it shares . . . The power also of the Decad. Without this, all things are unlimited, obscure and indiscernible.

One must study the activities and the essence of Number in accordance with the power existing in the Decad (Ten-ness); for it (the Decad) is great, complete, all-achieving, and the origin of divine and human life and its Leader; it shares . . . The power also of the Decad. Without this, all things are unlimited, obscure and indiscernible.

For the nature of Number is the cause of recognition, able to give guidance and teaching to every man in what is puzzling and unknown. For none of existing things would be clear to anyone, either in themselves or in their relationship to one another, unless there existed Number and its essence. But in fact Number, fitting all things into the soul through sense-perception, makes them recognisable and comparable with one another as is provided by the nature of the Gnômôn, 1 in that Number gives them body and divides the different relationships of things, whether they be Non-Limited or Limiting, into their separate groups.

And you may see the nature of Number and its power at work not only in supernatural and divine existences but also in all human activities and words everywhere, both throughout all technical production and also in music.

The nature of Number and Harmony admits of no Falsehood; for this is unrelated to them. Falsehood and Envy belong to the nature of the Non-Limited and the Unintelligent and the Irrational.

Falsehood can in no way breathe on Number; for Falsehood is inimical and hostile to its nature, whereas Truth is related to and in close natural union with the race of Number.

The bodies (physical Elements) of the Sphere are five: the Fire in the Sphere, and the Water, and Earth, and Air, and, fifth, the vehicle(?) 2 of the Sphere.

(The Four Elements of the rational animal are: Brain, Heart, Navel, Genital Organ). The Head is the seat of the Mind, the

The ancient theologians and seers also bear witness that because of certain punishments the soul is yoked to the body and buried in it as in a tomb.

(Socrates to Cebes in the 'Phaedo': The theory that we are in a sort of watch-tower which we must not desert is difficult; but I agree that we are one of the possessions of the gods). 1

There are certain thoughts which are stronger than ourselves.

Attributed to a prose work called 'Bacchae'

(Stobaeus: paraphrase). The universe is one, and it began to come into being from the centre, and from the centre upwards at the same intervals (of distance) as those below. For the parts above from the centre are in inverse relationship to those below; for the centre is to what is below as it is to what is above, and so with all the rest; for both stand in the same relationship to the centre, except in so far as their positions are reversed.

(Stobaeus: quotation on the Sun has fallen out).

(Proclus: Plato, Pythagorean doctrine and Philolaus in the 'Bacchae' teach theology through mathematical figures).

Doubtful fragments

(The Number Seven is like) Nikê (i.e. Athena), the Motherless and Virgin . . . for it is ruler and teacher of all things; it is God, One ever-existing, stable, unmoving, itself like to itself, different from the rest.

The Dyad (Two-ness) is the Bride of Cronos.

21. (Quoted by Stobaeus from a work attributed to Philolaus entitled 'On the Soul'). Therefore it (the universe) endures both indestructible and indomitable for endless time. For neither within it will there be found any cause more powerful than itself, nor outside it any cause able to destroy it. But this universe was from eternity and will endure till eternity, One, steered by One, akin to it and most powerful and unsurpassable. The universe, being one and continuous and inspired by the breath of Nature and carried round, has also the beginning of motion and change from its first beginning. And part of it is unchanging and part changing. And the unchanging part has its bounds from the Soul which encloses the Whole as far as the moon; and the changing part from the moon to the earth. But since the moving part circles from everlasting to everlasting, and the part that is moved is disposed in whatever way the moving part carries it, it follows necessarily that one is ever in motion and the other ever passive. And the one is wholly the dwelling (?) of Mind and Soul, and the other of Becoming and Change; and the one is first in power and superior, and the other is second and inferior. But that which is made of both these, namely the ever-running (circling) Divine and the ever-changing Mortal, is the universe.

It is well that the universe should be an everlasting activity of God and Becoming, according to the attendance (on God) of changeable Nature. And he remains One for ever in the same position and condition, but particular existences are coming into being and perishing in vast numbers. And they, when given over to destruction, yet keep their natures and forms, and in birth they reproduce the same form as their father and creator who gave them birth . . .

22. (Claudianus Mamertus (5th century A.D.) purporting to quote from the third volume of a treatise of Philolaus 'On Rhythm and Metre').

The soul is joined to the body through Number and the immortal and likewise incorporeal Harmony . . . The body is loved by the soul, because without it it cannot use the senses. When the soul has been separated from the body by death, it lives an incorporeal existence in the world.

23. (Iamblichus) Number is the ruling and self-created bond which maintains the everlasting stability of the contents of the universe.

Philolaus of Croton was a Greek philosopher and mathematician who lived in the late 5th century BCE. He was a student of Pythagoras and is best known for his contributions to Pythagorean philosophy, particularly his theory of the universe, which posited that the earth, moon, sun, and planets all revolve around a central "fire" at the center of the universe.

Philolaus was also interested in mathematics and made important contributions to the development of Pythagorean mathematics. He is credited with introducing the idea of the duality of numbers, which asserts that for every odd number there is a corresponding even number and vice versa.

Philolaus is considered an important figure in the history of philosophy and science, and his ideas had a profound influence on the development of Western thought.

In addition to his contributions to Pythagorean philosophy and mathematics, Philolaus is also known for his ideas about the nature of the soul and the afterlife. He believed that the soul was immortal and that after death it would be reincarnated into a new body. He also believed that the universe was governed by mathematical laws and that numbers were the key to understanding the world.

Philolaus was a prominent figure in the Pythagorean school, which was a philosophical and religious movement that emphasized the importance of mathematics and music in understanding the nature of the universe. He was known for his work on the harmony of the spheres, which held that the movement of the planets and stars was governed by mathematical principles and that this movement produced a kind of cosmic music.

Unfortunately, many of Philolaus' writings have been lost over time, and much of what we know about him comes from the writings of later philosophers who were influenced by his ideas. Despite this, Philolaus remains an important figure in the history of philosophy and science, and his contributions to Pythagorean thought continue to be studied and admired by scholars today.

Known fragments from "On Nature" by Philolaus of Croton:

1. Fragment 1: "The All is limited, both in respect to the world which is now visible and in respect to the time in which it came to be."

2. Fragment 2: "The nature of things is twofold, bodies and incorporeal things, of which the former are visible and the latter invisible."

3. Fragment 3: "The first principle of things is the Monas, or the One, from which all things proceed."

4. Fragment 4: "The universe is ordered according to harmonia, or the principles of harmony."

5. Fragment 5: "The universe is divided into two hemispheres, one visible and one invisible."

6. Fragment 6: "The earth is a sphere, and the center of the universe."

7. Fragment 7: "The earth, as a sphere, rotates around a central fire, which is the source of all light and heat."

8. Fragment 8: "The soul is immortal, and after death it is reincarnated into a new body."

9. Fragment 9: "All things are governed by mathematical principles, and numbers are the key to understanding the universe."