Part 36

i. 6; 987 b 22. But that the one is the real essence of things, and not something else with unity as an attribute, he affirms, agreeing with the Pythagoreans; and in harmony with them he affirms that numbers are the principles of being for other things. But it is peculiar to him that instead of a single infinite he posits a double infinite, an infinite of greatness and of littleness; and it is also peculiar to him that he separates numbers from things that are seen, while they say that numbers are the things themselves, and do not interpose mathematical objects between them. This separation of the one and numbers from things, in contrast with the position of the Pythagoreans, and the introduction of ideas, are the consequence of his investigation by concepts.

i. 8; 989 b 32-990 a 32. Those, however, who carry on their investigation with reference to all things, and divide things into what are perceived and what are not perceived by sense, evidently examine both classes, so one must delay a little longer over what they say. They speak correctly and incorrectly in reference to the questions now before us. Now those who are called Pythagoreans use principles and elements yet stranger than those of the physicists, in that they do not take them from the sphere of sense, for mathematical objects are without motion, except in the case of astronomy. Still, they discourse about everything in nature and study it; they construct the heaven, they observe what happens in its parts †and their states and motions†; they apply to these their first principles and causes, as though they agreed entirely with the other physicists that being is only what is perceptible and what that which is called heaven includes. But their causes and first principles, they say, are such as to lead up to the higher parts of reality, and are in harmony with this rather than with the doctrines of nature. In what manner motion will take place when finite and infinite, odd and even, are the only underlying realities, they do not say; nor how it is possible for genesis and destruction to take place without motion and change, or for the heavenly bodies to revolve. Farther, if one grant to them that greatness arises from these principles, or if this could be proved, nevertheless, how will it be that some bodies are light and some heavy? For their postulates and statements apply no more to mathematical objects than to things of sense; accordingly they have said nothing at all about fire or earth or any such objects, because I think they have no distinctive doctrine about things of sense. Farther, how is it necessary to assume that number and states of number are the causes of what is in the heavens and what is taking place there from the beginning and now, and that there is no other number than that out of which the world is composed? For when opinion and opportune time are at a certain point in the heavens, and a little farther up or down are injustice and judgment or a mixture of them, and they bring forward as proof that each one of these is number, and the result then is that at this place there is already a multitude of compounded quantities because those states of number have each their place—is this number in heaven the same which it is necessary to assume that each of these things is, or is it something different? Plato says it is different; still, he thinks that both these things and the causes of them are numbers; but the one class are ideal causes, and the others are sense causes.

ii. 1; 996 a 4. And the most difficult and perplexing question of all is whether unity and being are not, as Plato and the Pythagoreans say, something different from things but their very essence, or whether the underlying substance is something different, friendship, as Empedokles says, or as another says, fire, or water, or air.

ii. 4; 1001 a 9. Plato and the Pythagoreans assert that neither being nor yet unity is something different from things, but that it is the very nature of them, as though essence itself consisted in unity and existence.

1036 b 17. So it turns out that many things of which the forms appear different have one form, as the Pythagoreans discovered; and one can say that there is one form for everything, and the others are not forms; and thus all things will be one.

ix. 2; 1053 b 11. Whether the one itself is a sort of essence, as first the Pythagoreans and later Plato affirmed....

xi. 7; 1072 b 31. And they are wrong who assume, as do the Pythagoreans and Speusippos, that the most beautiful and the best is not in the first principle, because the first principles of plants and animals are indeed causes; for that which is beautiful and perfect is in what comes from these first principles.

xii. 4; 1078 b 21. The Pythagoreans (before Demokritos) only defined a few things, the concepts of which they reduced to numbers, as for instance opportunity or justice or marriage....

xii. 6; 1080 b 16. The Pythagoreans say that there is but one number, the mathematical, but things of sense are not separated from this, for they are composed of it; indeed, they construct the whole heaven out of numbers, but not out of unit numbers, for they assume that the unities have quantity; but how the first unity was so constituted as to have quantity, they seem at a loss to say. b 31. All, as many as regard the one as the element and first principle of things, except the Pythagoreans, assert that numbers are based on the unit; but the Pythagoreans assert, as has been remarked, that numbers have quantity.

xii. 8; 1083 b 9. The Pythagorean standpoint has on the one hand fewer difficulties than those that have been discussed, but it has new difficulties of its own. The fact that they do not regard number as separate, removes many of the contradictions; but it is impossible that bodies should consist of numbers, and that this number should be mathematical. Nor is it true that indivisible elements have quantity; but, granted that they have this quality of indivisibility, the units have no quantity; for how can quantity be composed of indivisible elements? but arithmetical number consists of units. But these say that things are number; at least, they adapt their speculations to such bodies as consist of elements which are numbers.

xiii. 3; 1090 a 20. On the other hand the Pythagoreans, because they see many qualities of numbers in bodies perceived by sense, regard objects as numbers, not as separate numbers, but as derived from numbers. And why? Because the qualities of numbers exist in harmony both in the heaven and in many other things. But for those who hold that number is mathematical only, it is impossible on the basis of their hypothesis to say any such thing; and it has already been remarked that there can be no science of these numbers. But we say, as above, that there is a science of numbers. Evidently the mathematical does not exist apart by itself, for in that case its qualities could not exist in bodies. In such a matter the Pythagoreans are restrained by nothing; when, however, they construct out of numbers physical bodies—out of numbers that have neither weight nor lightness, bodies that have weight and lightness—they seem to be speaking about another heaven and other bodies than those perceived by sense.

Eth. i. 4; 1096 b 5. And the Pythagoreans seem to speak more persuasively about it, putting the unity in the co-ordination of good things.

ii. 5; 1106 b 29. The evil partakes of the nature of the infinite, the good of the finite, as the Pythagoreans conjectured.

v. 8; 1132 b 21. Reciprocity seems to some to be absolutely just, as the Pythagoreans say; for these defined the just as that which is reciprocal to another.

Mor. i. 1; 1118 a 11. First Pythagoras attempted to speak concerning virtue, but he did not speak correctly; for bringing virtues into correspondence with numbers, he did not make any distinct.

PYTHAGORAS AND THE PYTHAGOREANS: PASSAGES IN THE DOXOGRAPHISTS.

Aet. Plac. i. 3; Dox. 280. And again from another starting-point, Pythagoras, son of Mnesarchos, a Samian, who was the first to call this matter by the name of philosophy, assumed as first principles the numbers and the symmetries existing in them, which he calls harmonies, and the elements compounded of both, that are called geometrical. And again he includes the monad and the undefined dyad among the first principles; and for him one of the first principles tends toward the creative and form-giving cause, which is intelligence, that is god, and the other tends toward the passive and material cause, which is the visible universe. And he says that the starting-point of number is the decad; for all Greeks and all barbarians count as far as ten, and when they get as far as this they return to the monad. And again, he says, the power of the ten is in the four and the tetrad. And the reason is this: if any one †returning† from the monad adds the numbers in a series as far as the four, he will fill out the number ten (i.e. 1 + 2 + 3 + 4 = 10); but if he goes beyond the number of the tetrad, he will exceed the ten. Just as if one should add one and two and should add to these three and four, he will fill out the number ten; so that according to the monad number is in the ten, but potentially in the four. Wherefore the Pythagoreans were wont to speak as though the greatest oath were the tetrad: ‘By him that transmitted to our soul the tetraktys, which has the spring and root of ever-flowing nature.’ And our soul, he says, is composed of the tetrad; for it is intelligence, understanding, opinion, sense, from which things come every art and science, and we ourselves become reasoning beings. The monad, however, is intelligence, for intelligence sees according to the monad. As for example, men are made up of many parts, and part by part they are devoid of sense and comprehension and experience, yet we perceive that man as one alone, whom no being resembles, possesses these qualities; and we perceive that a horse is one, but part by part it is without experience. For these are all forms and classes according to monads. Wherefore, assigning this limit with reference to each one of these, they speak of a reasoning being and a neighing being. On this account then the monad is intelligence by which we perceive these things. And the undefined dyad is science; fittingly, for all proof and all persuasion is part of science, and farther every syllogism brings together what is questioned out of some things that are agreed upon, and easily proves something else; and science is the comprehension of these things, wherefore it would be the dyad. And opinion as the result of comprehending them is the triad; fittingly, for opinion has to do with many things; and the triad is quantity, as ‘The thrice-blessed Danaoi.’ On this account then he includes the triad.... And their sect is called Italic because Pythagoras taught in Italy, for he removed from Samos, his fatherland, because of dissatisfaction with the tyranny of Polykrates.

Aet. i. 7; Dox. 302. Pythagoras held that one of the first principles, the monad, is god and the good, which is the origin of the One, and is itself intelligence; but the undefined dyad is a divinity and the bad, surrounding which is the mass of matter. i. 8; 307. Divine spirits [δαίμονες] are psychical beings; and heroes are souls separated from bodies, good heroes are good souls, bad heroes are bad souls. i. 9; 307. The followers of Thales and Pythagoras and the Stoics held that matter is variable and changeable and transformable and in a state of flux, the whole through the whole. i. 10; 309. Pythagoras asserted that the so-called forms and ideas exist in numbers and their harmonies, and in what are called geometrical objects, apart from bodies. i. 11; 310. Pythagoras and Aristotle asserted that the first causes are immaterial, but that other causes involve a union or contact with material substance [so that the world is material]. i. 14; 312. The followers of Pythagoras held that the universe is a sphere according to the form of the four elements; but the highest fire alone is conical. i. 15; 314. The Pythagoreans call colour the manifestation of matter. i. 16; 314. Bodies are subject to change of condition, and are divisible to infinity. i. 18; 316. (After quotation from Arist. Phys. iv. 4; 212 a 20) And in his first book on the philosophy of Pythagoras he writes that the heaven is one, and that time and wind and the void which always defines the places of each thing, are introduced from the infinite. And among other things he says that place is the immovable limit of what surrounds the world, or that in which bodies abide and are moved; and that it is full when it surrounds body on every side, and empty when it has absolutely nothing in itself. Accordingly it is necessary for place to exist, and body; and it is never empty except only from the standpoint of thought, for the nature of it in perpetuity is destructive of the interrelation of things and of the combination of bodies; and motions arise according to place of bodies that surround and oppose each other; and no infiniteness is lacking, either of quantity or of extent. i. 20; 318. Pythagoras said that time is the sphere of what surrounds the world. i. 21; 318. Pythagoras, Plato: Motion is a certain otherness or difference in matter. [This is the common limit of all motion.] i. 24; 320. Pythagoras and all that assume that matter is subject to change assert that genesis and destruction in an absolute sense take place; for from change of the elements and modification and separation of them there take place juxtaposition and mixture, and intermingling and melting together.

Aet. Plac. ii. 1; 327. Pythagoras first named the circumference of all things the universe by reason of the order in it. ii. 4; 330. Pythagoras, Plato, and the Stoics held that the universe is brought into being by god. And it is perishable so far as its nature is concerned, for it is perceived by sense, and therefore material; it will not however be destroyed in accordance with the foreknowledge and plan of god. ii. 6; 334. Pythagoras: The universe is made from five solid figures, which are called also mathematical; of these he says that earth has arisen 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. 9; 338. The followers of Pythagoras hold that there is a void outside the universe into which the universe breathes forth, and from which it breathes in. ii. 10; 339. Pythagoras, Plato, Aristotle: The right hand side of the universe is the eastern part from which comes the beginning of motion, and the left hand side is the west. They say the universe has neither height nor depth, in which statement height means distance from below upwards, and depth from above downwards. For none of the distances thus described exist for the universe, inasmuch as it is disposed around the middle of itself, from which it extends toward the all, and with reference to which it is the same on every side. ii. 12; 340. Thales, Pythagoras, and their followers: The sphere of the whole heaven is divided into five circles, which they call zones; the first of these is called the arctic zone and is ever visible; the second the summer solstice, the third the equinoctial, the fourth the winter solstice, and fifth the antarctic zone, which is invisible. And the ecliptic called the zodiac in the three middle ones is projected to touch the three middle ones. And the meridian crosses all these from the north to the opposite quarter at right angles. It is said that Pythagoras was the first to recognise the slant of the zodiacal circle which Oenopides of Chios appropriated as his own discovery. ii. 13; 343. Herakleides and the Pythagoreans asserted that each world [κόσμος] of the stars is air and aether surrounding earth in the infinite aether. And these doctrines are brought out in the Orphic writings, for they construct each world of the stars. ii. 22; 352. The Pythagoreans: The sun is spherical. ii. 23; 353. Plato, Pythagoras, Aristotle: The solstices lie along the slant of the zodiacal circle, through which the sun goes along the zodiac, and with the accompaniment of the tropic circles; and all these things also the globe shows. ii. 24; 354. An eclipse takes place when the moon comes past. ii. 25; 357. Pythagoras: The moon is a mirror-like body. i. 29; 360. Some of the Pythagoreans (according to the Aristotelian account and the statement of Philip the Opuntian) said that an eclipse of the moon takes place, sometimes by the interposition of the earth, sometimes by the interposition of the counter-earth [ἀντίχθων]. But it seems to some more recent thinkers that it takes place by a spreading of the flame little by little as it is gradually kindled, until it gives the complete full moon, and again, in like manner, it grows less until the conjunction, when it is completely extinguished. ii. 30; 361. Some of the Pythagoreans, among them Philolaos, said that the earthy appearance of the moon is due to its being inhabited by animals and by plants, like those on our earth, only greater and more beautiful; for the animals on it are fifteen times as powerful, not having any sort of excrement, and their day is fifteen times as long as ours. But others said that the outward appearance in the moon is a reflection on the other side of the inflamed circle of the sea that is on our earth. ii. 32; 364. Some regard the greater year ... as the sixty year period, among whom are Oenopides and Pythagoras.

Aet. Plac. iii. 1; Dox. 364. Some of the Pythagoreans said that the milky way is the burning of a star that fell from its own foundation, setting on fire the region through which it passed in a circle, as Phaethon was burned. And others say that the course of the sun arose in this manner at the first. And certain ones say that the appearance of the sun is like a mirror reflecting its rays toward the heaven, and therefore it happens at times to reflect its rays on the rainbow in the clouds.

Aet. iii. 2; 366. Some of the followers of Pythagoras say that the comet is one of the stars that are not always shining, but emit their light periodically through a certain definite time; but others say that it is the reflection of our vision into the sun, like reflected images. iii. 14; 378. Pythagoras: The earth, after the analogy of the sphere of the all, is divided into five zones, arctic, antarctic, summer, winter, and equinoctial; of these the middle one he defines to be the middle of the earth, called for this very reason the torrid zone; but the inhabited one [the one between the arctic and the torrid zones] being well-tempered....

Aet. iv. 2; Dox. 386. Pythagoras holds that number moves itself, and he takes number as an equivalent for intelligence. iv. 4; 389. Pythagoras, Plato: According to a superficial account the soul is of two parts, the one possessing, the other lacking, reason; but according to close and exact examination, of three parts; for the unreasoning part they divide into the emotions and the desires. (Theodor. v. 20); Dox. 390. The successors of Pythagoras saying that body is a mixture of five elements (for they ranked the aether as a fifth along with the four) held that the powers of the soul are of the same number as these. And these they name intelligence and wisdom and understanding and opinion and sense-perception. iv. 5; 391. Pythagoras: The principle of life is about the heart, but the principle of reason and intelligence is about the head. iv. 5; 392. Pythagoras et al.: The intelligence enters from without. iv. 7; 392. Pythagoras, Plato: The soul is imperishable. iv. 9; 396. Pythagoras et al.: The sense-perceptions are deceptive. iv. 9; 397. Pythagoras, Plato: Each of the sensations is pure, proceeding from each single element. With reference to vision, it was of the nature of aether; hearing, of the nature of wind; smell, of the nature of fire; taste, of the nature of moisture; touch, of the nature of earth. iv. 14; 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 to the shoulder. iv. 20; 409. Pythagoras, Plato, Aristotle: Sound is immaterial. For it is not air, but it is the form about the air and the appearance [ἐπιφανεία] after some sort of percussion which becomes sound; and every appearance is immaterial; for it moves with bodies, but is itself absolutely immaterial; as in the case of a bent rod the surface-appearance suffers no change, but the matter is what is bent.

Aet. Plac. v. 1; 415. Pythagoras did not admit the sacrificial part alone (of augury). v. 3; 417. Pythagoras: The seed is foam of the best part of the blood, a secretion from the nourishment, like blood and marrow. v. 4; 417. Pythagoras, Plato, Aristotle: The power of seed is immaterial, like intelligence, the moving power; but the matter that is poured forth is material. v. 20; 432. Pythagoras, Plato: The souls of animals called unreasoning are reasonable, not however with active reasoning powers, because of an imperfect mixture of the bodies and because they do not have the power of speech, as in the case of apes and dogs; for these have intelligence but not the power of speech.

Ar. Did. Ep. Fr. 32; Dox. 467. Apollodoros in the second book Concerning the gods: It is the Pythagorean opinion that the morning and the evening star are the same.

Theophr. Phys. Op. Fr. 17; Dox. 492. Favorinus says that he (Pythagoras) was the first to call the heavens a universe and the earth round [στρογγύλην].

Cic. de Deor. Nat. i. 11; Philod. piet. Fr. c 4 b; Dox. 533. For Pythagoras, who held that soul is extended through all the nature of things and mingled with them, and that from this our souls are taken, did not see that god would be separated and torn apart by the separation of human souls; and when souls are wretched, as might happen to many, then part of god would be wretched; a thing which could not happen.

Hippol. Phil. 2; Dox. 555. There is a second philosophy not far distant from the same time, of which Pythagoras, whom some call a Samian, was the first representative. And this they call the Italian philosophy because Pythagoras fled the rule of Polykrates over the Samians and settled in a city of Italy where he spent his life. The successive leaders of this sect shared the same spirit. And he in his studies of nature mingled astronomy and geometry and music <and arithmetic>. And thus he asserted that god is a monad, and examining the nature of number with especial care, he said that the universe produces melody and is put together with harmony, and he first proved the motion of the seven stars to be rhythm and melody. And in wonder at the structure of the universe, he decreed that at first his disciples should be silent, as it were mystae who were coming into the order of the all; then when he thought they had sufficient education in the principles of truth, and had sought wisdom sufficiently in regard to stars and in regard to nature, he pronounced them pure and then bade them speak. He separated his disciples into two groups, and called one esoteric, and the other exoteric. To the former he entrusted the more perfect sciences, to the latter the more moderate. And he dealt with magic, as they say, and himself discovered the art of physiognomy. Postulating both numbers and measures he was wont to say that the first principle of arithmetic embraced philosophy by combination, after the following manner:

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 female number, and the same is called by the arithmeticians even. Thirdly the triad is 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.