(being continued from 13/10/15)

**[1] FRAGMENTS FROM THE DOXOGRAPHERS**

** i. 9; Dox.307** The followers of Thales and Pythagoras and the Stoics held that matter is variable and changeable and transformable and is in state of flux, the whole through the whole. [p.308]

** i10; Dox.309** Pythagoras asserted that the so-called forms and ideas exist in numbers and their harmonies, and in what are geometrical objects, apart from bodies. [p.309]

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

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

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

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

** ii.5. Dox.357.** Pythagoras: The moon is a mirror-like body. [p.309]

** Aetius, Plac. iv. 2; Dox. 386.** Pythagoras holds that number moves itself, and he takes number as an equivalent for intelligence. [p.310]

** iv.14;Dox.405** The followers of Pythagoras and of the mathematicians on reflections of vision: for vision moves directly as it were against the bronze (of a mirror) and meeting with a firm, smooth surface, it is turned and bent back on itself meeting some such experience as when the arm is extended and then bent back at the shoulder. [p.311]

** v.4;Dox 417** Pythagoras, Plato, Aristotle: The power of seed is immaterial, like intelligence, the moving power, but the matter that is poured forth is material. [p.311]

** Hippol., Phil,. 2. Dox. 355 **….[ According to Pythagoras]… Number is the first principle, a thing which is undefined, incomprehensible, having in itself all numbers which could reach infinity in amount. And the first principle of numbers is in substance the first Monad, which is a male monad, begetting as a father all other numbers. Secondly, the Dyad is a female number, and the same is called by the arithmeticians even. Thirdly, the Triad is a male number; this the arithmeticians have been wont to call odd. Finally, the Tetrad is a female number, and the same is called even because it is female. …. Pythagoras said this sacred Tektractys is: `

**.’**

*the spring having the roots of ever-flowing nature*…. the four parts of the Decad, this perfect number, are called number, monad, power and cube. And the interweavings and minglings of these in the origin of growth are what naturally completes nascent number; for when a power of a power; and a cube is multiplied on a cube, it is the power of a cube; and when a cube is multiplied on a cube, the cube of a cube; thus all numbers, from which arise the genesis of what arises, are seven: number, monad, power, cube, power of a power, power of a cube, and cube of a cube. [p.312]

[** Contact with Zoroaster**] …and he says the universe exists in accordance with musical harmony, so the sun also makes an harmonious period. And concerning the things that arise from the earth and the universe they say Zaratas spoke as follows: ` There are two divinities, one of the heavens and the other of the earth; the one of earth produces things from the earth, and it is water; and the divinity of the heavens is fire with a portion of air, warm, and cold; wherefore he says that none of these things will destroy or even pollute the soul, for these are the essence of all things. [p.313]

** Pythagoras perished** in a conflagation with his disciples in Croton in Italy. And it was the custom when one became a disciple to burn one’s property and leave one’s money under a seal with Pythagoras, and one remained in silence sometimes three years, and sometimes five years, and studied.

[** Other Contacts**] ..and immediately on being released from this one mingled with the others and continued as a disciple and made one’s home with them; otherwise one took one’s money and was sent off. The esoteric class were called Pythagoreans, and the others Pythagoristians. And those disciples who escaped the conflagation were Lysis and Archippus and Zalmoxis the slave of Pythagoras who is said to have taught the Pythagorean philosophy to the Druids among the Celts. It is said that Pythagoras learned numbers and measures from the Egyptians. [p.313]

**[2] 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. [p. 168]

**4 (Nicomachus, Arith. Intr., 2. 509)** . …. it would not be possible that any of the things that exist and that are known to 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 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 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 wholes, plus one semitone. [p.168]

**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 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 a minor half-tone. He supposes that this sharp contain 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 unity plays the part of the point, 3 the odd first 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 a comma, and that 27 unities form an entire 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, in the interval whose eight-ninths relation exceeds the sum of two sharps, namely, the sum of two semitones. The schisma is half the comma, the diaschisma is half the sharp, namely, of the minor semitone.

**8 (Claudanus Mamertus, De Statu Animæ, 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. [p. 169]

**9 (Nicomachus, Arithm. Intr., 2. p. 72).** Some, in this following Philolaus, think that this kind of 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. [p.169]

**B. (Cassidorus, Exp. in Ps., 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. [p. 169]

**10** **A. (Stobæus, Eclog. Physic., 1. 15. 7. p.360)**. The world is single and it came into be 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 other parts; in fact, in respect to the center, each one of the opposite points is identical, unless the whole be moved.

**B. (Stobæus, Eclog. Physic., 1. 21. 1. p.468)**. The prime composite, the One placed in the center of the sphere, is called Hestia. [p.170]

(to be continued)