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PASSAGES REFERRING TO THE PYTHAGOREANS FROM THE DOXOGRAPHERS

TRANSLATED BY ARTHUR FAIRBANKS

THE WRITINGS of the doxographers, the Vetusta Placita of Aetius, the Placita Philosophorum attributed to Plutarch, and so on, ultimately derive from a work of Theophrastus, Physikon Doxon. "Opinions of the Natural Philosophers." In this work Theophrastus compiled information on the doctrines of different philosophers by subject, serving to compare and contrast them with one another.

The fragments of the doxographers were collected together and published by Hermann Diels as Doxographi Graeci, Berlin, 1879, and the corresponding numeration of Diels' edition is reproduced at the beginning of each fragment. These fragments relating to the Pythagoreans, like the preceding passages from Plato and Aristotle, are translated by Arthur Fairbanks and reproduced from his The First Philosophers of Greece: An Edition and Translation of the Remaining Fragments of the Pre-Sokratic Philosophers, Together with a Translation of the More Important Accounts of their Opinions Contained in the Early Epitomes of their Works, New York, Charles Scribner's Sons, 1898.

FRAGMENTS FROM THE DOXOGRAPHERS

Aetius, Plac. i. 3; Dox. 280. And again from another starting-point, Pythagoras, son of Mnesarchus, 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 Indefinite 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 ten is in four and the tetrad. And the reason is this: if anyone from the monad adds the numbers in a series as far as 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 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 [actually] 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,
The spring and root of ever-flowing nature.

And our soul, he says, is composed of the Tetrad, for this is intelligence, understanding, opinion, and 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, possessing these qualities; and we perceive that a horse is one, but part by part it is without experience. For these are 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 Indefinite Dyad is fittingly science, for all proof and persuasion is part of science, and further 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 fittingly the Triad, 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, having left Samos, his fatherland, of dissatisfied with the tyranny of Polycrates.

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 Indefinite Dyad is a daimon and bad, surrounding which is the mass of matter.

i. 8; Dox. 307. Divine spirits (daimones) are psychical beings, and heroes are souls separated from bodies; good heroes are good souls, bad heroes are bad souls.

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 a state of flux, the whole through the whole.

i. 10; Dox. 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; Dox. 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; Dox. 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; Dox. 314. The Pythagoreans call color the manifestation of matter.

i. 16; Dox. 314. Bodies are subject to change of condition, and are divisible to infinity.

i. 18; Dox. 316. (After a quotation from Aristotle, Physics, 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 from the standpoint of thought, for the nature of it in perpetuity destroys the interrelation of things and the combination of bodies; motions arise according to the place of bodies that surround and oppose each other, and no infiniteness is lacking, either of quality or of extent.

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.

i. 24; Dox. 320. Pythagoras, and all that assume matter is subject to change, assert that genesis and destruction in an absolute sense take place, for from change of the elements, modification and separation of them there takes place juxtaposition and mixture, and intermingling and melting together.

Aetius, Plac. ii. l; Dox. 327. Pythagoras first named the circumference of all things the kosmos by reason of the order in it.

ii. 4; Dox. 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; Dox. 334. Pythagoras: The universe is made from five solid figures which are also called 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; Dox. 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; Dox. 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 distance thus described exists 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; Dox. 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 the 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 recognize the slant of the zodiacal circle which Oenopides of Chios appropriated as his own discovery.

ii. 13; Dox. 343. Heracleides and the Pythagoreans asserted that each world 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 [likewise] construct each world of the stars.

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

ii. 23; Dox. 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 tropical circles, and all these things the globe also shows.

ii. 24; Dox. 354. An eclipse takes place when the moon comes past.

ii. 25; Dox. 357. Pythagoras: The moon is a mirror-like body.

ii. 29; Dox. 360. Some of the Pythagoreans (according to the Aristotelian account and Philip of Opus) said that an eclipse of the moon takes place sometimes by the interposition of the earth, sometimes by the interposition of the counterearth (antichthon). 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; Dox. 361. Some of the Pythagoreans, among them Philolaus, 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; Dox. 364. Some regard the greater year...as the sixty year period, among whom are Oenopides and Pythagoras.

Aetius, 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.

iii. 2; Dox. 366. Some of the followers of Pythagoras say that a comet is one of the stars which are not always shining, but which 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; Dox. 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, the inhabited one [the one between the arctic and the torrid zones] being well-tempered...

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

iv. 4, Dox. 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 emotion and the desires.

Theodor. v. 20; Dox. 390. The successors of Pythagoras saying that the body is a mixture of five elements (for they ranked 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 named intelligence and wisdom and understanding and opinion and sense-perception.

Aetius, Plac. iv. 5; Dox. 391. Pythagoras: The principle of life is about the heart, but the principle of reason and intelligence is about the head.

iv. 5; Dox. 392. Pythagoras, et al.: The intelligence enters from without.

iv. 7; Dox. 392. Pythagoras, Plato: The soul is imperishable.

iv. 9; Dox. 396. Pythagoras, et al.: The sense-perceptions are deceptive.

iv. 9; Dox. 397. Pythagoras, Plato: Each of the sensations is pure, proceeding from each single element. With reference to vision, it is 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; 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 to the shoulder.

iv. 20; Dox. 409. Pythagoras, Plato, Aristotle: Sound is immaterial. For it is not air, but it is the form about the air and the appearance (epiphaneia) 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.

Aetius, Plac. v. 1; Dox. 415. Pythagoras did not admit the sacrificial part alone (of augury).

v. 3; Dox. 417. Pythagoras: Sperm is foam of the best part of the blood, a secretion from the nourishment, like blood and marrow.

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.

v. 20; Dox. 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 is the case of apes and dogs, for these have intelligence but not the power of speech.

Ar. Did. Ep. Fr. 32; Dox. 467. Apollodorus 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 kosmos and the earth spherical.

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 Polycrates over the Samians and settled in an Italian city 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 kosmos 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 if they 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 to 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 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.

All numbers, then, taken by classes are four -- but number is undefined in reference to class -- of which is composed the perfect number, the Decad. For the series one, two, three and four becomes ten, and its own name is kept in its essence by each of the numbers. Pythagoras said that this sacred Tetraktys is "the spring having the roots of ever-flowing nature" in itself, and from this numbers have their first principle. For the eleven and the twelve and the rest derive from the ten the first principle of their being. 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 is multiplied upon itself, it is the power of a power; and when a power 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.

He said that the soul is immortal, and that it changes from one body to another; so he was wont to say that he himself had been born before the Trojan war as Aethalides, and at the time of the Trojan war as Euphorbus, and after that as Hermontimus of Samos, then as Pyrrhos of Delos, fifth as Pythagoras. And Diodorus of Eretria and Aristoxenus the musician say that Pythagoras had come unto Zaratas of Chaldaea [i.e., Zoroaster]; and he set forth that in his view there were from the beginning two causes of things: father and mother. The father is light and the mother darkness; and the parts of light are warm, dry, light, swift; and of darkness are cold, moist, heavy, slow; and of all these the universe is composed, of male and female. And he says that 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 the 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. And it is said that Zaratas forbade men to eat beans because he said that at the beginning and composition of all things when the earth was still a whole, the bean arose. And he says that the proof of this is that if one chews a bean to a pulp and exposes it to the sun for a certain time (for the sun will affect it quickly), it gives off the odor of human seed. And he says that there is another and clearer proof: if when a bean is in flower we were to take the bean and its flower, and putting it into a pitcher moisten it and then bury it in the earth, and after a few days dig it up again, we should see in the first place that it had the form of a womb, and examining it closely we should find the head of a child growing with it.

Pythagoras perished in a conflagration with his disciples in Croton in Italy. And it was the custom when one became a disciple to burn one's property and to leave one's money under a seal with Pythagoras, and one remained in silence sometimes three years, and sometimes five years, and studied. 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 of the disciples who escaped the conflagration 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. Astonished at the wisdom of the priests, which was deserving of belief and full of fancies and difficult to grasp, he imitated it and himself also taught his disciples to be silent, and obliged the student to remain quietly in rooms underneath the earth.

Epiph. Pro. i; Dox. 587. Pythagoras laid down the doctrine of the Monad and of foreknowledge and the prohibition on sacrificing to the Gods then believed in, and he bade men not to partake of beings that had life, and to refrain from wine. And he drew a line between the things from the moon upwards, calling these immortal, and those below, which he called mortal; and he taught the transmigration of souls from bodies into bodies even as far as animals and beasts. And he used to teach his followers to observe silence for a period of five years. Finally he named himself a God. [1]

Epiph. Haer. iii. 8; Dox. 390. Pythagoras the Samian, son of Mnesarchus, said that the Monad is God, and that nothing has been brought into being apart from this. He was wont to say that wise men ought not to sacrifice animals to the Gods, nor yet to eat what had life, nor beans, nor to drink wine. And he was wont to say that all things from the moon downward were subject to change, while from the moon upward they were not. And he said that the soul goes at death into other animals. And he bade his disciples to keep silence for a period of five years, and finally he named himself a God.

Herm. I.G.P. 16; Dox. 655. Others then from the ancient tribe, Pythagoras and his fellow tribesmen, revered and taciturn, transmitted other dogmas to me as mysteries, and this is the great and unspeakable ipse dixit: the Monad is the first principle of all things. From its forms and from numbers the elements arose. And he declared that the number, form and measure of each of these is somehow as follows: Fire is composed of twenty-four right-angled triangles, surrounded by four equilaterals. And each equilateral consists of six right- angled triangles, whence they compare it to the pyramid. Air is composed of forty-eight triangles, surrounded by eight equilaterals. And it is compared to the octahedron, which is surrounded by eight equilateral triangles, each of which is separated into six right-angled triangles so as to become forty-eight in all. And water is composed of one hundred and twenty triangles, surrounded by twenty equilaterals, and it is compared to the icosahedron, which is composed of one hundred and twenty equilateral triangle. And aither is composed of twelve equilateral pentagons, and is like a dodecahedron. And earth is composed of forty-eight triangles, and is surrounded by six equilateral tetragons, and it is like a cube. For the cube is surrounded by six tetragons, each of which is separated into eight triangles, so that they become in all forty-eight.


FIGURE 16. THE REGULAR SOLIDS. The regular solids, also known as the Platonic solids, were first described by Plato in his Timaeus. Plato identified the dodecahedron with the cosmic sphere (later identified with aither), and the four other solids with the four elements. Each one of the elemental "molecules" is constructed out of the triangular "atoms" shown below. The five regular solids are the only polyhedra that can be constructed out of the same regular polygons. The archetypal ratios and geometries with which they are associated underlie the structure and divisions of three-dimensional space.

DODECAHEDRON: Aither / 12 Sides
TETRAHEDRON: Fire / 4 Sides
CUBE: Earth / 6 Sides
OCTAHEDRON: Air / 8 Sides
ICOSAHEDRON : Water / 20 Sides



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Notes:

1. While Pythagoras may have thought of himself as having some type of special relationship with the God Apollo, there is no reason to believe that he ever thought of himself as being a God. In fact, other church fathers attributed to Pythagoras a statement which challenged any man who thought himself a God to create a universe.

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APPENDIX I: HOW MANY TETRAKTYS ARE THERE?

BY THEON OF SMYRNA

THE IMPORTANCE OF THE QUATERNARY obtained by addition (that is to say 1 + 2 + 3 + 4) is great in music because all the consonances are found in it. But it is not only for this reason that all Pythagoreans hold it in highest esteem: it is also because it seems to outline the entire nature of the universe. It is for this reason that the formula of their oath was: "I swear by the one who has bestowed the Tetraktys to the coming generations, source of eternal nature, into our souls." The one who bestowed it was Pythagoras, and it has been said that the Tetraktys appears indeed to have been discovered by him.

The first quaternary is the one of which we've just spoken: it is formed by addition of the first four numbers.

The second is formed by multiplication, of even and odd numbers, starting from unity. Of these numbers, unity is the first because, as we have said, it is the principle of all the even numbers, the odd numbers and of all the odd-even numbers, and its essence is simple. Next comes three numbers from the odd as well as the even series. They allow for the unification of odd and even because numbers are not only odd or even. For this reason, in multiplication, two quaternaries are taken, one even, the other odd; the even in double ratio, the first of the even numbers being 2 which comes from unity doubled; the odd in triple ratio, the first of the odd numbers being the number 3 which arise from unity being tripled, so that unity is odd and even simultaneously and belongs to both. The second number in the even and double [series] is 2 and in the odd and triple is 3. The third of the order of even numbers is 4, and in the odd series, 9. The fourth among the even numbers is 8, and among the odd numbers, 27.


FIGURE 17. THE PLATONIC LAMBDA

The ratios of the most perfect consonances are found in these numbers; even the tone is included. However unity contains the principle of ratio, of limit and of point. The second numbers, 2 and 3, have the side ratio, being prime, incomposite numbers, and measured only by the unit, and are consequently linear numbers. The third terms, 4 and 9, have the power of the squared surface, being equally equal (that is to say square numbers). The fourth terms, 8 and 27, have the power of the cubic solid, being equally equal equally (that is to say, cubic numbers). In this way, by virtue of the numbers from this tetraktys, growth proceeds from the limit and the point up to the solid. In fact, after the limit and the point comes the side, then the surface and finally the solid. It is with these numbers that Plato, in the Timaeus, constitutes the [world] soul. [1] The last of these seven numbers is equal to the sum of all the preceding, as we have 1+2+3+4+8+9=27.

There are then two quaternaries of numbers, one which is made by addition, the other by multiplication; and these quaternaries encompass the musical, geometric and arithmetic ratios of which the harmony of the universe is composed.

The third quaternary is that which, following the same proportion, embraces the nature of all magnitudes, for the place taken by unity, in the preceding quaternary, is that of the point in this one; and that of the numbers 2 and 3, having lateral (or linear) power, is here that of the line, through its double form, straight or circular, the straight line corresponding to the even number because it terminates at two points (the line and circle are given as examples here), and the circular to the odd, because it is composed of a single line without terminus.

And what, in the preceding quaternary, are the numbers 4 and 9, having the power of the surface, the two types of surface, the planar and the curved, are so (surface) in this one. Finally, what, in the preceding are the numbers 8 and 27, which have the power of the cube and of which one is even and the other odd, is constituted by the solid in this one. There are two kinds of solids, one with a curved surface, like the sphere or the cylinder, the other with a plane surface, such as the cube and the pyramid. This is the third tetraktys then, the one having the property of constituting any magnitude, through the point, the line, the surface and the solid.

The fourth quaternary is that of the simple bodies, fire, air, water and earth, and it offers the same proportion as the quaternary of numbers. The place occupied by unity in the quaternary of numbers is taken by fire in this one, air corresponds to the number 2, water to the number 3, earth to the number 4; such is indeed the nature of the elements according to their fineness or density, in such a way that fire is to air as 1 is to 2, to water as 1 is to 3, and to earth as 1 is to 4. The other relationships are also equal (that is to say, that air is to water as 2 is to 3, and so forth for the others).

The fifth quaternary is that of the shapes of simple bodies, for the pyramid is the figure of fire, the octahedron the figure of air, the icosahedron the figure of water and the cube the figure of earth.

The sixth is that of the created things, the seed being analogous to unity and the point. A growth in length is analogous to the number 2 and the line, and a growth in width is analogous to the number 3 and to the surface, and finally a growth in thickness is analogous to the number 4 and to the solid.

The seventh quaternary is that of societies. Man is principle and is thus unity. The family corresponds to the number 2, the village to the number 3 and the city to the number 4; for these are the elements which comprise the nation.

All of these quaternaries are material and perceptible.

The eighth contains faculties by which we are able to form judgment on the preceding, and which are its intellectual part, namely: thought, science, opinion and feeling. And certainly thought, in its essence, must be assimilated to unity; science is the number 2, because it is the science of all things; opinion is like the number 3, because it is something between science and ignorance; and finally feeling is like the number 4 because it is quadruple, the sense of touch being common to all, all the senses being motivated through contact.

The ninth quaternary is that which composes the living things, body and soul, the soul having three parts, the rational, the emotional and the willful; the fourth part is the body in which the soul resides.

The tenth quaternary is that of the of the seasons of the year, through the succession of which all things take birth, that is, spring, summer, autumn and winter.

The eleventh is that of the ages: childhood, adolescence, maturity and old age.

There are thus eleven quaternaries. The first is that of the numbers which are formed by addition, the second is that of the numbers formed by multiplication, the fourth is that of magnitudes, the fifth is that of simple bodies, the sixth is that of created things, the seventh is that of societies, the eighth is that of the faculties of judgment, the ninth is that of the living things, the tenth is that of the seasons, and the eleventh is that of the ages. They are proportional to one another, since what is unity in the first and the second quaternary, the point is in the third, fire is in fourth, the pyramid in the fifth, the seed in the sixth, man in the seventh, thought in the eighth, and so forth with the others following the same proportion.

Thus the first quaternary is 1, 2, 3, 4. The second is unity, the side, the square, the cube. The third is the point, the line, the surface, the solid. The fourth is fire, air, water, earth. The fifth is the pyramid, the octahedron, the icosahedron, the cube. The sixth is the seed, the length, the width, the height. The seventh is man, the family, the village, the city. The eighth is thought, science, opinion, sense. The ninth is the rational, the emotional and the willful parts of the soul, and the body. The tenth is spring, summer, autumn, and winter. The eleventh is childhood, adolescence, maturity and old age. And the perfect world which results from these quaternaries is geometrically, harmonically and arithmetically arranged, containing in power the entire nature of number, every magnitude and every body, whether simple or composite. It is perfect because everything is part of it, and it is itself a part of nothing else. This is why the Pythagoreans used the oath whose formula we have reported, and through which all things are assimilated to number.

From Theon of Smyrna: Mathematics Useful for Understanding Plato, Chapter 38. Translated by Robert and Deborah Lawlor. San Diego, Wizards Bookshelf, 1979. Reproduced with permission of the publisher.

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Notes:

1. Plato, Timaeus 36BC.

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APPENDIX II: THE PYTHAGOREAN TITLES OF THE FIRST TEN NUMBERS

FROM THE THEOLOGY OF NUMBERS BY IAMBLICHUS

TRANSLATED BY DAVID R. FIDELER

PLUTARCH AND PLOTINUS inform us that the Pythagoreans called the One Apollo because of its lack of multiplicity -- this is both a clever pun and a revealing statement, for a-pollon in Greek means literally "not of many."

Such, then, was the Greek style of "theological arithmetic." This form of number symbolism became quite popular in late antiquity and much of it was transmitted by Christian writers through Medieval times. The symbolism finds its basis in the Pythagorean observation that the primary numbers represent far more than quantitative signs: each one of the primary numbers is a qualitative, archetypal essence, possessing a distinct, living personality. This personality can be directly intuited by studying the archetypal manifestations of these principles in the realms arithmetic (number in itself), geometry (number in space) and harmonics (number in time).

The following list represents a compilation of the titles from the anonymous Theology of Arithmetic which was based closely on a work by Iamblichus. Since the first printing of this volume, however, a complete, fully annotated translation of The Theology of Arithmetic has appeared, which represents the first translation of this work into any European language (The Theology of Arithmetic, translated by Robin Waterfield, Phanes Press, Grand Rapids, 1988.) In the text, explanations are given for the various appellations. For more on arithmology also see book three of Thomas Taylor's Theoretic Arithmetic of the Pythagoreans and the section on number symbolism in Thomas Stanley's Pythagoras.


THE MONAD

Instrument of Truth
Obscure
Not-Many
A Chariot
Male-Female
Immutable Truth and Invulnerable Destiny
A Seed
Fabricator (demiurge)
True Happiness (eudaimonia)
Zeus
Life
God
The Equality in Increase and |Decrease
Memory
A Ship
Essence (ousia)
The Innkeeper (pandokeus), "that which takes in all"
The Pattern or Model (paradeigma)
The Moulder
Prometheus
The First (Proteus)
Darkness
Blending
Commixture
Harmony (symphonia)
Order (taxis)
Materia
A Friend
Infinite Expanse (chaos)
Space-Producer


THE DYAD

Inequality
Indefinite (aoristos)
The Unlimited (apeiron)
Without Form or Figure
Growth
Birth
Judgment
Appearance
Anguish
The Each of Two
Falling Short, Defect
Erato
Equal
Isis
Movement
The Ratio (logos) in Proportion (analogia)
Revolution
Distance
Impulse
Excess
The Thing with Another
Rhea (the wife of Kronos, but also "flow")
Selene
Combination
That Which is To Be Endured; Misery, Distress
Boldness, Audacity (tolma)
Matter
Obstinacy
Nature


THE TRIAD

Proportion (analogia)
Harmonia
Marriage
Knowledge (gnosis)
Peace
Every Thing
Hecate
Good Counsel
Piety
The Mean Between Two Extremes
Oneness of Mind
The All
Perfection
Friendship
Purpose


THE TETRAD

Nature of Change
Righteousness
Hercules
Holding the Key of Nature


THE PENTAD

Alteration
Immortal
Androgyny
Lack of Strife
Aphrodite
Boubastia (named after the Egyptian divinity Boubastis)
Wedding
Marriage
Double
Manifesting Justice
Justice
Demigod
Nemesis
Pallas
Five-Fold
Forethought
Light


THE HEXAD

Resembling Justice
The Thunder-Stone
Amphitrite (Poseidon's wife; a verbal pun: on both sides [amphis] three [trias])
Male-Female
Marriage
Finest of All
In Two Measures
Form of Forms
Peace
Far-Shooting (name of Apollo)
Thaleia
Kosmos
Possessing Wholeness
Cure-All (panacea)
Perfection
Three-Fold
Health
Reconciling


THE HEPTAD

The Forager (epithet of Athena)
Athena
Citadel (akropolis)
Reaper
Hard to Subdue Defence
Due Measure (kairos)
Virgin (parthenos)
Revered Seven (septas + sebomai = heptas)
Bringing to Completion
(Telesphorus)
Fortune, Fate
Preserving


THE OCTAD

Untimely Born
Steadfast
Seat or Abode
Euterpe
Cadmia
Mother
All Harmonious


THE ENNEAD

Brother and Consort of Zeus
Helios
Absence of Strife
Far-Working (epithet of Apollo)
Hera
Hephaestus
Maiden (kore)
Of the Kouretes
Assimilation
Oneness of Mind
Horizon (because it limits the series of units before returning to the Decad)
Crossing or Passage
Prometheus
Consort and Brother
Perfection
Bringing to Perfection (Telesphorus)
Terpsichore
Hyperion
Oceanus


THE DECAD

Eternity (aeon)
Untiring
Necessity
Atlas
Fate
Helios
God
Key-Holding
Kosmos
Strength
Memory
Ourania
Heaven
All
All Perfect
Faith
Phanes

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FIGURE 18. THE DIVINE MONOCHORD. This particular monochord is tuned in the key of G, while the examples [below] and in the introduction use the key of C. The three top notes on this monochord are incorrectly placed.

APPENDIX III: THE FORMATION AND RATIOS OF THE PYTHAGOREAN SCALE
BY DAVID R. FIDELER


FIGURE 19. THE RATIOS OF THE PYTHAGOREAN SCALE

AS NOTED in the introductory essay, the structure of the musical scale possesses a great deal of significance in Pythagorean thought as it is an excellent example of the principle of mathematical harmonia at work. In the case of the scale, the "opposites" of the high (2) and the low (1) -- the two extremes of the octave -- are united in one continuum of tonal relationships through the use of a variety of forms of proportion which actively mediate between these two extremes.

The best way to understand the mathematical principles of harmonic mediation involves actually charting out and playing out the ratios of the scale on the monochord. In constructing a monochord, it is best to make it as long as possible, perhaps in the region of 4-5 feet, as that makes it easier to differentiate between the harmonic nodal points at the high end of the spectrum (see fig. 6).

It is useful at first to play out the harmonic overtone series. Measure the exact length of the string and then mark off the overtone intervals: 1/2 the string length, 1/3 the string length, 1/4 the string length, etc. It is possible to play out the overtone series without the use of the bridge; simply pluck the string about 1 inch from either end while simultaneously touching the nodal point with the other hand. It will be noted that there is an inverse relationship between the vibrational frequency of the tone and the string length. This is also illustrated in the above chart: hence a tone with a vibration of 2 is associated with a string division of .5 or 1/2. It is also useful at this point to play out the harmonic "Tetraktys," or the perfect consonances: 1:2 (octave), 2:3 (perfect fifth), and 3:4 (perfect fourth). Listen carefully to these ratios and reflect on the fact that you are actually hearing the relationships between these primary whole numbers.

To "tune" the monochord to the ratios of the Pythagorean scale use the string length ratios in the above chart, multiplying these ratios by the length of the string. Mark off these intervals, along with the corresponding notes, on the sounding board as they are carefully measured out.

Having marked out the Pythagorean scale, it might be useful at this point to review the material in the introductory essay relating to the harmonic proportion and then to play out these relations:

1) Play out the relationship of the octave (1:2). These are the two tonal extremes which must be united.

2) Play out the arithmetic mean linking together the extremes: C-G-c, or 6-9-12. This is the perfect fifth, the strongest musical relationship (2:3).

3) Play out the harmonic mean linking together the two extremes: C-F-c, or 6-8-12. This is the perfect fourth, the next strongest musical relationship (3:4).

4) Now play out the harmonic or musical proportion which is the basis of the musical scale: C-F-G-c, or 6:8::9::12. Play this out as a continued proportion and then the individual parts. Play out the two perfect fifths 6:9 and 8:12. Play out the two perfect fourths 6:8 and 9:12. Then play out the whole tone 8:9.

5) Having played out the harmonic foundation of the scale, now "fill in" the remaining 8:9 whole tone intervals. Play out C-D, D-E, G-A, and A-B. Along with F-G, these are all in the 8:9 ratio.

6) Play out the ratio of the leimma or the semitone: E-F and B-c. The leimma is the relationship between the perfect fourth and three whole tones.

7) Finally play out the entire scale: C-D-E-F-G-A-B-c. Through the use of arithmetic, harmonic and geometric proportion the two extremes have been successfully united.

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APPENDIX IV: A SUMMARY OF PYTHAGOREAN MATHEMATICAL DISCOVERIES

BY SIR THOMAS HEATH

NOT ONLY did the early Pythagoreans make many contributions in the realm of philosophy, but their mathematical studies laid the foundation for the development of Greek geometry, and many portions of Euclid's Elements can be traced back to mathematical discoveries of the Pythagorean school.

This listing of early Pythagorean mathematical discoveries is excerpted from Thomas Heath's History of Greek Mathematics, vol. I, pp, 166-169.

A SUMMARY OF PYTHAGOREAN MATHEMATICAL DISCOVERIES

1. They were acquainted with the properties of parallel lines, which they used for the purpose of establishing by a general proof the proposition that the sum of the three angles of any triangle is equal to two right angles. This latter proposition they again used to establish the well-known theorems about The sums of the exterior and interior angles, respectively, of any polygon.

2. They originated the subject of equivalent areas, the transformation of an area of one form into another of different form and, in particular, the whole method of application of areas, constituting a geometrical algebra, whereby they effect the equivalent of the algebraical processes of addition, subtraction, multiplication, division, squaring, extraction of the square root, and finally the complete solution to the mixed quadratic equation x2 ± pxq = 0 so far as its roots are real. Expressed in terms of Euclid, this means the whole content of Book I. 35-48 and Book II. The method of application of areas is one of the most fundamental in the whole of later Greek geometry; it takes its place by the side of the powerful method of proportion; moreover, it is the starting point of Apollonius' theory of conics, and the three fundamental terms, parabole, ellipsis, and hyperbole used to describe the three separate problems in 'application' were actually employed by Apollonius to denote the three conics, names which, of course, are those which we use to-day. Nor was the use of the geometrical algebra for solving numerical problems unknown to the Pythagoreans; this is proved by the fact that the theorems of Eucl. II. 9, 10 were invented for the purpose of finding successive integral solutions of the indeterminate equations

2x2 - y2 = ± 1

3. They had a theory of proportion pretty fully developed. We know nothing of the form in which it was expounded; all we know is that it took no account of incommensurable magnitudes. Hence we conclude that it was a numerical theory, a theory on the same lines as that contained in Book VII of Euclid's Elements.

They were aware of the properties of similar figures. This is clear from the fact that they must be assumed to have solved the problem, which was, according to Plutarch, attributed to Pythagoras himself, of describing a figure which shall be similar to one given figure and equal in area to another given figure. This implies a knowledge of the proposition that similar figures (triangles or polygons) are to one another in the duplicate ratio of corresponding sides (Eucl. VI. 19, 20). As the problem is solved in Eucl. VI. 25, we assume that, subject to the qualification that their theorems about similarity, &c., were only established of figures in which corresponding elements are commensurable, they had theorems corresponding to a great part of Eucl., Book VI.

Again, they knew how to cut a straight line in extreme and mean ratio (Eucl. VI. 30); [1] this problem was presumably solved by the method used in Eucl. II. 11, rather than by that of Eucl. VI. 30, which depends on the solution of a problem in the application of areas more general than the methods of Book II enable us to solve, the problem namely of Eucl. VI. 29.

4. They had discovered, or were aware of the existence of, the five regular solids. These they may have constructed empirically by putting together squares, equilateral triangles, and pentagons. This implies that they could construct a regular pentagon and, as this construction depends upon the construction of an isosceles triangle in which each of the base angles is double of the vertical angle, and this again on the cutting of a line in extreme and mean ratio, we may fairly assume that this was the way in which the construction of the regular pentagon was actually evolved. It would follow that the solution of problems by analysis was already practised by the Pythagoreans, notwithstanding that the discovery of the analytical method is attributed by Proclus to Plato. As the particular construction is practically given in Eucl. IV. 10, 11, we may assume that the content of Eucl. IV was also partly Pythagorean.

5. They discovered the existence of the irrational in the sense that they proved the incommensurability of the diagonal of a square with reference to its side; in other words, they proved the irrationality of √2. At; a proof of this is referred to by Aristotle in terms which correspond to the method used in a proposition interpolated in Euclid, Book X, we may conclude that this proof is ancient, and therefore that it was probably the proof used by the discoverers of the proposition. The method is to prove that, if the diagonal of a square is commensurable with the side, then the same number must be both odd and even; here then we probably have of early Pythagorean use of the method of reductio ad absurdum.

Not only did the Pythagoreans discover the irrationality of √2; they showed, as we have seen, how to approximate as closely as we please to its numerical value. After the discovery of this one case of irrationality, it would be obvious that propositions theretofore proven by means of the numerical theory of proportion, which was inapplicable to incommensurable magnitudes, were only partially proved. Accordingly, pending the discovery of a theory of proportion applicable to incommensurable as well as commensurable magnitudes, there would be an inducement to substitute, where possible, for proofs employing the theory of proportion other proofs independent of that theory. This substitution is carried rather far in Euclid, Books I-IV; it does not follow that the Pythagoreans remodelled their proofs to the same extent as Euclid felt bound to do.

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Notes:

1. The extreme and mean division of the line is an important mathematical and geometrical ratio which underlies various universal forms. This is the so-called "Divine Proportion," "Golden Section," or Phi ratio. On the properties and significance of this principle see Ghyka, The Geometry of Art and Life, and other titles on sacred geometry listed in the bibliography.

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GLOSSARY OF SELECT PYTHAGOREAN TERMS

Analogia -- Literally, "through proportion." Hence, continued geometrical proportion or ratio. See logos.

Apeiron -- Boundless; Unlimited; Infinite; Indefinite. One of the Pythagorean first principles in the Table of Opposites. See peras and Indefinite Dyad.

Harmonia -- A "joint" or "fitting together;" hence, the musical scale comprised within the octave. Also, music per se; philosophically, the principle of Union, opposite Strife. Also, a Pythagorean name of the number Three, because a third element must be introduced to mediate between or join together two separate principles or numerical values.

Indefinite Dyad -- Plato's term for the Pythagorean principle of Apeiron, as contrasted with the One, the principle of Limit.

Kosmos -- "Order." Also, "ornament." First applied to the universe by Pythagoras, hence cosmos means world-order. Also, a Pythagorean name of the numbers Six and Ten.

Logos -- Usually translated "Word" or "Reason." In the mathematical and Pythagorean sense, the same as the Latin ratio, i.e. "proportion;" hence also, a principle of mediation. Can also mean "principle;" the plural logoi can be translated as "principles," "reasons" or "causes," or (mathematical) "ratios."

Mean or median -- The middle term in a mathematical proportion which links two extreme terms together in harmonia. The three most important are the Arithmetic, Harmonic, and Geometric means, which underlie the structure of the musical scale in Pythagorean tuning. In the following equations, the two extremes are A and C, and the mean term is B.

Arithmetic Mean: B = (A +C) / 2
Harmonic Mean: B = 2AC / A + C
Geometric Mean: B = √A x C

Monochord (kanon) -- A one-stringed musical instrument with a movable bridge used for dividing the string at any length. The monochord is used to demonstrate the harmonic overtone series and the principles on which the musical scale is based.

Peras -- The principle of Limit or Boundary. The opposite of Apeiron, the Unlimited.

Symphonia -- Literally, "sounding together." "Harmony," agreement or concord. The term applies to the perfect intervals or consonance of the octave, fifth, and fourth, par excellence. This is the modern meaning of the word "harmony," but not the ancient one.

Tetraktys -- (from tetras, four). "Fourness." Also, the first four numbers, especially when arranged in an equilateral triangle, the sum of which is the number Ten. Hence also, the Decad.

The Tetraktys symbolizes the perfection of Number and the elements which comprise it. The Tetraktys also contains the symphonic ratios which make possible the musical scale, i.e., 1:2, the octave; 2:3, the perfect fifth; and 3:4, the perfect fourth.

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A SELECT BIBLIOGRAPHY OF WORKS RELATING TO PYTHAGORAS AND PYTHAGOREAN PHILOSOPHY

COMPILED BY DAVID R. FIDELER AND JOSCELYN GODWIN

WHILE THE FOLLOWING BIBLIOGRAPHY of 400 titles cannot be considered as constituting the final word on writings which relate to Pythagorean studies, it does include all the important works that we are familiar with, and some works that are less frequently cited.

To make this bibliography more useful, the listing has been divided into several categories: Pythagorean Texts, Secondary Sources, Classical Philosophy, Mathematics, Music, Astronomy, Medieval and Renaissance, Sacred Geometry, and Whole Systems. A few words of explanation have been added underneath each category heading, and titles are arranged alphabetically by author under each category. The major problem with this format is that certain works could fall into more than one category. Despite this potential shortcoming, it was decided that the virtues of this arrangement outweigh any potential drawbacks.

For individuals who are beginning a study of Pythagorean thought, the best starting point is probably the lengthy and very well written section on the Pythagoreans to be found in W.K.C. Guthrie's History of Greek Philosophy, vol. 1. While his conclusions are not universally accepted, another excellent study is Cornford's article "Science and Mysticism in the Pythagorean Tradition." In terms of general histories of Greek philosophy, the accounts of Pythagorean thought given by John Burnet in Early Greek Philosophy and John Robinson in An Introduction to Early Greek Philosophy are better than most. The topic of Neopythagorean thought in the late Hellenistic period has yet to receive the full scale treatment that it deserves, but good accounts of Neopythagorean philosophers can be found in Dillon's The Middle Platonists. No account of Pythagorean scholarship would be complete without mentioning the massive, though perhaps hypercritical, study of Walter Burkert, Lore and Science in Ancient Pythagoreanism. Two other significant and delightful studies, which approach the topic from somewhat different angles, are Vogel's Pythagoras and Early Pythagoreanism: An Interpretation of Neglected Evidence on the Philosopher Pythagoras and Heninger's Touches of Sweet Harmony: Pythagorean Cosmology and Renaissance Poetics. Finally, Holger Thesleff has made very important contributions to Pythagorean studies in two volumes, An Introduction to the Pythagorean Writings of the Hellenistic Period and The Pythagorean Writings of the Hellenistic Period. In the latter volume Thesleff has collected together and edited the Greek texts of the Hellenistic Pythagorica.

PYTHAGOREAN TEXTS

The writings in this section are individual Pythagorean or Neopythagorean writings. If these writings appear in this volume, that fact is noted and the page number is given.

Anonymous. Accounts of Pythagorean thought in Sextus Empiricus, Vol. I, 429-39; Vol. 2, 49-57; Vol. 3, 331-61, Loeb Classical Library, Harvard University Press, 1933.

Anonymous. The Golden Verses of the Pythagoreans. Brook, Surrey, Shrine of Wisdom, n.d.

Anonymous. The Golden Verses of Pythagoras. In this volume, 163-65.

Anonymous. The Life of Pythagoras preserved by Photius. In this volume, 137-40.

Anonymous. Pythagorean Symbols or Maxims. In this volume, 159-61.

Apollonius of Tyana. The Letters of Apollonius of Tyana: A Critical Text with Prolegomena, Translation and Commentary by R. J. Penella, Leiden, E.J, Brill, 1979.

Archytas. The Fragments of Archytas. In Chaignet, Pythagore et la philosophie pythagoricienne, Paris, 1874, vol. 1, 256-331; in this volume, 177-201.

Aristoxenus of Tarentum. Apothegms. In Taylor, Political Fragments, 65; in this volume, 243.

Boeckh. Philolaus des Pythagoreer's Lehren, nebst den Bruchstucken seiner Werke. Berlin, 1819.

Brown, Hellen Ann. Philosophorum Pythagoreorum collectionis specimen. Diss., University of Chicago, 1944. (Greek texts.)

Callicratidas. On the Felicity of Families. In Taylor, Political Fragments, 50-57; in this volume, 235-37.

Cardini, Maria Timpanaro, ed. Pitagorici: Testimonianze e frammenti. 3 vols. Florence, La Nuova Italia, 1958-64. (Greek texts.)

Chaignet, A.E. Pythagore et la philosophie pythagoricienne. 2 vols. Paris, Librairie Academique, 1874. (Contains the fragments of Philolaus and Archytas).

Charondas the Catanean. Preface to the Laws of Charondas the Catanean. In Taylor, Political Fragments, 38-45; in this volume, 231-33.

Clinias. A Fragment of Clinias. In Taylor, Life of Pythagoras, 167; in this volume, 265.

Crito. On Prudence and Prosperity. In Taylor, Life of Pythagoras, 177-79; in this volume, 251-52.

Delatte, A. La Vie de Pythagore de Diogene Laerce: edition critique avec introduction et commentaire. Brussels, Lamertin, 1922.

Demophilus. "The Pythagoric Sentences of Demophilus," in Sallust on the Gods and the World. Trans. by Thomas Taylor. (1793) Los Angeles, Philosophical Research Society, 1976.

Diehl, E., and Young, David, eds. Theognis, ps. -Pythagoras, ps. -Phocylides. Leipzig, Teubner, 1961.

Diels, H. and Kranz, W. Die Fragmente der Vorsokratiker. 6th ed., 3 vols. Berlin, Weidmann, 1951-2. (Greek texts, translated in Freeman, Ancilia to the Pre-Socratic Philosophers, Harvard University Press, 1983.)

Diogenes Laertius. Lives of the Eminent Philosophers. 2 vols. Trans. by R.D. Hicks. Harvard University Press, 1925.

Diogenes Laertius. Life of Pythagoras. In this volume, 141-56.

Diotogenes. Concerning a Kingdom. In Taylor, Political Fragments, 18-26; in this volume, 222-24.

Diotogenes. On Sanctity. In Taylor, Political Fragments. 10-11, 37-38; in this volume, 221.

Ecphantus the Crotonian. On a Kingdom. In Taylor, Political Fragments. 27-37; in this volume, 257-59.

Euryphamus. Concerning Human Life. In Taylor, Life of Pythagoras, 148-50; in this volume, 245-46.

Fabre d'Olivet, Antoine. Les Vers Dores de Pythagore. (1813) Paris, L'Age d'Homme, n.d.

Fabre d'Olivet, Antoine. The Golden Verses of Pythagoras. Trans. by N.L. Redfield. (1917) New York, Weiser, 1975.

Fairbanks, Arthur. The First Philosophers of Greece: An Edition and Translation of the Remaining Fragments of the Pre-Sokratic Philosophers. Together with a Translation of the More Important Accounts of Their Opinions Contained in the Early Epitomes of Their Works. New York, Charles Scribners, 1898.

Freeman, Kathleen. Ancilla to the Pre-Socratic Philosophers. (1948) Harvard University Press, 1983.

Hadas, Moses, and Smith, Morton. Heroes and Gods: Spiritual Biographies in Antiquity. New York, Harper & Row, 1965. (Contains a trans. of Porphyry's Life of Pythagoras.)

Hierocles. Ethical Fragments. In Taylor, Political Fragments. 71-115; in this volume, 275-86.

Hierocles. Commentary of Hierocles on the Golden Verses of Pythagoras. Translated by N. Rowe from the French version of Andre Dacier. (1907) London, Theosophical Publishing House, 1971.

Hipparchus. On Tranquility. In Taylor, Life of Pythagoras. 151-53; in this volume, 247-48.

Hippodamus the Thurian. On a Republic. In Taylor, Political Fragments. 1-10, 17; in this volume, 217-20.

Hippodamus the Thurian. On Felicity. In Taylor, Life of Pythagoras. 143-47; in this volume, 215-17.

Iamblichus. Iamblichi De vita Pythagorica liber; accedit epimetrum De Pythagorae Aureo carmine. St. Petersburg, 1884. (Greek text.)

Iamblichus. De Vita Pythagorica. Ed. by Deubner. Leipzig, Teubner, 1937. (Greek text.)

Iamblichus. De vita Pythagorica liber. Edited by August Nauck. (1884) Amsterdam, 1965.

Iamblichus. The Life of Pythagoras. Trans. by Thomas Taylor. (1818) London, I.M. Watkins, 1965; in this volume, 57-122.

Iamblichus. Exhortation to Philosophy. Trans. by Thomas M. Johnson. Grand Rapids. Phanes Press, 1988. (Part of Iamblichus' "Pythagorean Encyclopedia;" contains his commentary on the Pythagorean Maxims.)

Iamblichus. (Pseudo-Iamblichus) Theologumena Arithmeticae. Edited by V. de Falco. Leipzig, Teubner, 1922. (Greek text.)

Maddalena, A. I Pitagorici: raccolta delle testimonianze e dei frammenti pervenutici. Bari, 1954. (Italian trans. of texts in DK with essays and notes.)

Metopus. Concerning Virtue. In Taylor, life of Pythagoras, 164-66; in this volume, 249.

Ocellus Lucanus. On the Nature of the Universe. (1831) Trans. by Thomas Taylor. Los Angeles, Philosophical Research Society, 1976.

Ocellus Lucanus. On the Nature of the Universe. In this volume, 203-11.

Pempelus. On Parents. In Taylor, Political Fragments. 67-69; in this volume, 261.

Perictyone. On the Harmony of a Woman. In Taylor, Political Fragments. 57-65; in this volume, 239-41.

Philolaus. The Fragments of Philolaus. In Chaignet, Pythagore et la philosophie pythagoricienne. Paris, 1874, vol. 1, p. 226-54; in this volume, 167-75.

Philostratus. The life of Apollonius of Tyana. 2 vols. Trans. by F.C. Conybeare. Harvard University Press, 1912.

Phyntis. On Woman's Temperance. In Taylor, Political Fragments, 69-74; in this volume, 263-64.

Polus. On Justice. In Taylor, life of Pythagoras. 182; in this volume, 253.

Porphyrius. "Vita Pythagorae" in Porphyrii Opuscula Selecta, ed. A. Nauck, 2nd edn. (1886) Hildesheim, 0lms, 1963.

Porphyry. The Life of Pythagoras. In this volume, 123-35.

Sextus Pythagoreus. The Sentences of Sextus. Ed. and trans. by R. Edwards and R. Wild. Chico, Scholars Press, 1981.

Sextus. Select Sentences of Sextus the Pythagorean. In Taylor, Life of Pythagoras. 192-200; in this volume, 267-70.

Speusippus. Fragments from his work On Pythagorean Numbers in Thomas, Greek Mathematical Works, vol. I, Harvard University Press, 1939.

Sthenidas the Locrian. On a Kingdom. In Taylor, Political Fragments, 26-27, in this volume, 255.

Taylor, Thomas, trans. Political Fragments of Archytas. Charondas. Zaleucus and Pythagoreans. preserved by Stobaeus; and also Ethical Fragments of Hierocles, the celebrated Commentator on the Golden Pythagoric Verses. preserved by the same author. London, 1822.

Theages. On the Virtues. In Taylor, life of Pythagoras, 161-63, 168-73; in this volume, 225-28.

Thesleff, Holger. The Pythagorean Writings of the Hellenistic Period. Abo, Abo Akaderni, 1965. (A complete collection of the Hellenistic Pythagorean writings in the original Greek.)

Timaeus Locrus. De natura mundi et animae. Uberlieferung, Testimonia, Text und Ubersetzung von W. Marg. Editio maior. Leiden, E.J. Brill, 1972.

Timaeus Locrus. Uber die Natur des Kosmos und der Seele. Konunentiert von M. Baltes. Leiden, E.J. Brill, 1972.

Timaios of Locri. On the Nature of the World and the Soul. Trans. by T.H. Tobin. Chico, Scholars Press, 1985.

Timaeus of Locri. On the World and the Soul. In this volume, 287-96.

Zaleucus the Locrian. Preface to the Laws of Zaleucus the Locrian. In Taylor, Political Fragments, 46-50; in this volume, 229-30.

SECONDARY SOURCES

The writings in this section, while not actual Pythagorean writings, are all specific studies of Pythagoras, Pythagorean thought, or Pythagorean writings.

Balch, L. "The Neo-pythagorean Moralists and the New Testament," in H. Temporini and W. Haase (ed.), Aufstieg und Niedergang der roemischen Welt, Berlin, de Gruyter, 1983, Teil II, Band 26.

Bamford, Christopher. "Homage to Pythagoras," in Lindisfarne Letter 14.

Bindel, Ernst. Pythagoras. Stuttgart, Freies Geistesleben, 1962.

Bomer, F. Der Lateinische Neuplatonismus und Neupythagoreismus. Leipzig, Harrassowitz, 1936.

Boehm, F. De Symbolis Pythagoreis. Diss., Berlin, 1905.

Boyance, Pierre. "Note sur la Tetractys," L'Antiquite Classique 20 (1951), 421-5.

Boyance, Pierre. "Sur l'Abaris d'Heraclide le Pontique," Revue des etudes anciennes 36 (1934) 321-52.

Boyance, Pierre. "Sur la vie pythagoricienne," Revue des etudes grecques 52 (1939), 36-50.

Burkert, Walter. "Hellenistiche Pseudopythagorica," Philologus 105 (1961), 16-43, 226-246.

Burkert, W. Lore and Science in Ancient Pythagoreanism. Harvard University Press, 1972.

Burnet, John. "Pythagoras and Pythagoreanism," in Encyclopedia of Religion and Ethics, New York, 1919, vol. X, 520-530.

Bywater, J. "On the Fragments Attributed to Philolaus the Pythagorean," Journal of Philology 1 (1868), 20-53.

Cameron, Alister. The Pythagorean Background of the Theory of Recollection. Menasha, WI, George Banta, 1938.

Carcopino, Jerome. La basilique pythagoricienne de la Pone Majeure. Paris, L'artisan du livre, 1927.

Carcopino, Jerome. Aspects Mystiques de la Rome Paienne. Paris, L'Artisan du Livre, 1941.

Carcopino, Jerome. De Pythagore aux Apotres. Paris, Flammarion, 1956.

Casa, Adriana della. Nigidio Figulo. Rome, Ateneo, 1962.

Chaignet, A.E. Pythagore et la philosophie pythagoricienne. 2 vols. Paris, Librairie Academique, 1874.

Cornford, F.M. "The Earliest Pythagorean Cosmology," in his Plato and Parmenides. London, Paul, Trench, Trubner, 1939.

Cornford, F.M. "The Invention of Space," in Essays in Honor of Gilbert Murray. London, Allen and Unwin, 1936.

Cornford, Francis M. "Mysticism and Science in the Pythagorean Tradition," Classical Quarterly 16 (1922), 137-150; 17 (1923), 1-12.

Dacier, Andre. The life of Pythagoras. (Trans. from French, London, 1707.) York Beach, Samuel Weiser, 1981.

Delatte, Armand. Etudes sur la litterature pythagoricienne. Paris, Champion, 1915.

Delatte, Armand. Essai sur la politique pythagoricienne. Paris, Champion, 1922.

Dillon, John. The Middle Platonists. Cornell University Press, 1977. (Contains a good account of the Neopythagoreans.)

Eliade, Mircea. "Orpheus, Pythagoras, and the New Eschatology," chapter 22 in A History of Religious Ideas, vol. 2, University of Chicago Press, 1982.

Festugiere, A. "Les memoires pythagoriques cites par Alexandre Polyhistor," Revue des etudes grecques 58 (1945), 1-65.

Frank, Erich. Plato und die sogenannten Pythagoreer. Halle, 1923.

Freeman, Kathleen. Companion to the Pre-Socratic Philosophers: A Companion to Diels' Fragmente Der Vorsokratiker. Oxford University Press, 1946.

Fritz, Kurt von. "Pythagoras of Samos," in The Dictionary of Scientific Biography, New York, Charles Scribner's Sons, 1975, vol. 9, 214-25.

Fritz, Kurt von. Pythagorean Politics in Southern Italy: An Analysis of the Sources. (1940) New York, Octagon Books, 1977.

Fritz, K. von, Dorrie, H., and van der Waerden, B.L. "Pythagoras" and "Pythagoreer" in Pauly-Wissowa Realencyclopadie der classischen Altertumswissenschaft, vol. 27 (1963), and suppl. vol. 10 (1965).

Godwin, Joscelyn. "Pythagoreans, Today?", in Lindisfarne Letter 14.

Goettling, C.W. "Die Symbole des Pythagoras," Gesammelte Abhandlungen, vol. 1, Halle, 1851.

Goodenough, E.R. "A Nee-Pythagorean Source in Philo Judaeus." Yale Classical Studies
3 (1932), 117-64.

Gorman, Peter. Pythagoras, a life. London, Routledge & Kegan Paul, 1979.

Guthrie, W.K.C. A History of Greek Philosophy. Vol. 1: The Earlier Presocratics and The Pythagoreans. Cambridge University Press, 1971.

Haase, Rudolf. "Literatur zur Geschichte des harmonikalen Pythagoreismus" [558 entries] in Aufsatze zur harmonikalen Naturphilosophie, Graz, Akademische Druck-und Verlagsanstalt, 1974.

Heidel, W.A. "Peras and Apeiron in the Pythagorean Philosophy," Archiv fur Geschichte der Philosophie 14 (1901), 384-399.

Heidel, W.A. "Notes on Philolaus," American Journal of Philosophy 28 (1907), 77-81.

Jager, Hans. Die Quellen des Porphyrios in seiner Pythagoras-Biographie. Diss., Zurich, Chur, 1919.

Kahn, Charles. "Pythagorean Philosophy before Plato," in Morelatos, The Pre-Socratics: A Collection of Essays, Garden City, Anchor Press, 1974, 161-85.

Kucharski, Paul. "Les principes des pythagoriciens et la dyade de Platon," Archives de Philosophie 22 (1959), 385-431.

Kucharski, Paul. "Aux frontieres du P1atonisme et du Pythagorisme," Archives de Philosophie 19.1 (1955-56), 7-43.

Levy, Isidore. La legend de Pythagore de Grece en Palestine. Paris, Champion, 1927.

Levy, Isidore. Recherches sur les sources de la legend de Pythagoras. Paris, Leroux, 1926.

Lindisfarne Letter 14, "Homage to Pythagoras." West Stockbridge, MA, Lindisfarne Press, 1982.

Long, H.S. A Study of the Doctrine of Metempsychosis in Greece from Pythagoras to Plato. Diss., Princeton, 1948.

Mallinger, Jean. Pythagore et les mysteres. Paris, Nic1aus, 1944.

Minar, Edwin L., Jr. "Pythagorean Communism," Transactions of the American Philological Association 75 (1944), 34-46.

Minar, E.L. Early Pythagorean Politics in Practice and Theory. Baltimore, Waverly Press, 1942.

Morrison, J.S. "Pythagoras of Samos," Classical Quarterly 50 (1956), 135-156.

Morrison, J.S. "The Origins of Plato's Philosopher-Statesman," Classical Quarterly 52 (1958), 198-218.

Oppermann, Hans. "Eine Pythagoraslegend," Bonner Jahrbucher 130 (1925), 284-301.

Philip, J.A. "Aristotle's Monograph On the Pythagoreans," Transactions of the American Philological Association 94 (1963), 185-198.

Raine, Kathleen. "Blake, Yeats and Pythagoras," in Lindisfarne Letter 14.

Raven, J.E. Pythagoreans and Eleatics. Cambridge University Press, 1948.

Richardson, Hilda. "The Myth of Er (Plato, Republic, 616B)," Classical Quarterly 20 (1926), 113-34.

Ridgeway, William. "What Led Pythagoras to the Doctrine that the World was Built of Numbers?" Classical Review 10 (March 1896), 92-95.

Rivaud, A. "Platon et la politique pythagoricienne," Melanges Gustave Glotz, vol. 2. Paris, Presses Universitairs de France, 1932.

Rohde, Erwin. "Die Quellen des Iamblichus in seiner Biographie des Pythagoras," Rheinisches Museum fur Philologie 26 (1871), 554-6; 27 (1872), 23-61.

Rosenthal, F. "Some Pythagorean Documents Transmitted in Arabic," Orientalia 10 (1941), 104-15; 383-95.

Rougier, Louis. L' Origine astronomique de la croyance pythagoricienne en l'immortalite celeste des ames. Cairo, Institut francais d'Archeologie orientale, 1933.

Rougier, L. La Religion astrale des Pythagoriciens. Paris, Presses Universitairs de France, 1959.

Rutherford, Ward. Pythagoras: Lover of Wisdom. Aquarian Press, Wellingborough, 1984.
Santillana, G. de and Pitts, W. "Philolaus in Limbo: or, "What happened to the Pythagoreans?", Isis (42) 1951, 112-20.

Seltman, C.T. "The Problem of the First Italiote Coins," Numismatic Chronicle, 6th series, 9 (1949), 1-21.

Stanley, Thomas. "Pythagoras," from his History of Philosophy. (1687) Los Angeles, Philosophical Research Society, 1970.

Stapelton, H.E. "Ancient and Modern Aspects of Pythagoreanism, " Osiris 13 (1958), 12-53.

Stocks, J.L. "Plato and the Tripartite Soul," Mind, n.s., 24 (1915), 207-21.

Swanson, R. A. "Ovid's Pythagorean Essay," Classical Journal 54 (1958), 21-24.

Taran, L. Asclepius of Tralles: Commentary to Nicomachus' Introduction to Arithmetic. (Transactions of the American Philosophical Society, n.s., 59, 4.) Philadelphia, American Philosophical Society, 1969.

Taylor, A.E. "Two Pythagorean Philosophemes," Classical Review 40 (1926), 149-51.

Thesleff, Holger. An Introduction to the Pythagorean Writings of the Hellenistic Period. Abo, Abo Akademi, 1961.

Vogel, Cornelia 1. de. Pythagoras and Early Pythagoreanism: An Interpretation of Neglected Evidence on the Philosopher Pythagoras. Assen, Van Gorcum, 1966.

Waerden, B.L. van der. "Die Harmonielehre der Pythagoreer," Hermes 78 (1968), 163-199.

Watters, Hallie. The Pythagorean Way of Life with a Discussion of the Golden Verses. (Masters thesis.) Adyar, Theosophical Publishing House, 1926.

Wellmann, Max. "Eine pythagoreische Urkunde des IV. Jahrhunderts vor Christus," Hermes (1919), 225-48.

White, M. "The Duration of the Samian Tyranny," Journal of Hellenic Studies 74 (1954), 36-43.

Whittaker, John. "Epekeina nou kai ousias, " Vigilate Christianae 23 (1969), 91-104. (On Moderatus.)

Whittaker, John. "Neopythagoreanism and Negative Theology." Symbolae Osloenses 44 (1969), 109-25.

Whittaker, John. "Neopythagoreanism and the Transcendent Absolute," Symbolae Osloenses 48 (1973), 77-86.

Zeller, Eduard. A History of Greek Philosophy: From the Earliest Period to the Time of Socrates. Trans. by S.F. Alleyne. Vol. I: Pre-Socratic Philosophy. London, 1881.

GREEK PHILOSOPHY

These works on early classical philosophy and Greek religion discuss Pythagorean thought.

Armstrong, A.H., ed. Cambridge History of Later Greek and Early Medieval Philosophy. Cambridge University Press, 1967.

Aujoulat, Noel. Le Neo-Platonisme Alexandrin: Hierocles d 'Alexandrie. Leiden, E.J. Brill, 1987.

Baldry, H.C. "Embryological Analogies in Presocratic Cosmogony." Classical Quarterly 26 (1932), 27-34.

Booth, N.B. "Were Zeno's Arguments Directed Against the Pythagoreans?" Phronesis 2 (1957), 90-103.

Brommer, P. "De numeris idealibus," Mnemosyne 3.11 (1943), 263-295.

Brumbaugh, Robert S. Plato's Mathematical Imagination: The Mathematical Passages in the Dialogues and Their Interpretation. Bloomington, Indiana University Press, 1954.

Burnet, J. Early Greek Philosophy: Part I. Thales to Plato. London, Macmillan, 1914.

Burnet, John. Early Greek Philosophy. 4th ed. London, Macmillan 1930.

Capek, M. "The Theory of Eternal Recurrence," Journal of Philosophy (57) 1960, 289-96.

Copleston, S.J. A History of Philosophy: Greece and Rome. New York, Doubleday, 1962.

Cornford, F.M. "Mystery Religions and Pre-Socratic Philosophy," in The Cambridge Ancient History, Cambridge University Press, 1926, vol. 4, 522-78.

Cornford, F.M. Principium Sapientiae: The Origins of Greek Philosophical Thought. Cambridge University Press, 1952.

Cornford, F.M. Plato's Cosmology. London, Routledge, 1937.

Cornford, F.M. The Laws of Motion in Ancient Thought. Cambridge University Press, 1931.

Cornford, F.M. From Religion to Philosophy. London, Arnold, 1912

Critchlow, Keith. "The Platonic Tradition on the Nature of Proportion," in Lindisfarne Letter 10.

Gomperz, Theodor. Greek Thinkers. London, Murray, 1912.

Guthrie, W.K.C. "The Presocratic World-Picture," Harvard Theological Review, 45 (1952), 87-104.

Kirk, G.S., and Raven, J.E. The Presocratic Philosophers: A Critical History with a Selection of Texts. Cambridge University Press, 1957.

Merlan, Philip. From Platonism to Neoplatonism. The Hague, Nijhoff, 1960.

Onians, R.B. The Origins of European Thought about the Body, the Mind. the Soul. the World. Time and Fate. Cambridge University Press, 1951.

Porphyry. On Abstinence from Animal Food. Edited by E. Wynne-Tyson. London, Barnes & Noble, 1965.

Robinson, John. An Introduction to Early Greek Philosophy. New York, Houghton Mifflin, 1968.

Rohde, Erwin. Psyche: The Cult of Souls and Belief in Immortality Among the Greeks. London, Routledge, 1925.

Taran, L. Speusippus of Athens: A Critical Study with a Collection of the Related Texts and Commentary. Leiden, E.J. Brill, 1982.

Taylor, A.E. A Commentary on Plato's Timaeus. Oxford University Press, 1928.

Zeller, Eduard. Outlines of the History of Greek Philosophy. New York, Dover, 1980.

MATHEMATICS

These works contain information on the mathematical discoveries of the Pythagoreans. Also included are works dealing with Pythagorean arithmology and geometry.

Adam, James. The Nuptial Number of Plato. (1891) London, Kairos, 1985.

Allendy, Rene. Le Symbolisme des nombres. Paris, Charcornac, 1921.

Anatolius. (Edited by Heiberg.) Anatolius sur les db: premiers nombres. (On the Numbers Up to Ten.) Macon, Protat Freres, 1900. (Greek text.)

Butler, Christopher. Number Symbolism. New York, Barnes & Noble, 1970.

Fritz, Kurt von. "The Discovery of Incommensurability by Hippasos of Metapontum," Annals of Mathematics 46 (1945), 242-264.

Gardner, Martin. "Simple Proofs of the Pythagorean Theorem," Scientific American 211.4 (Oct. 1964), 118-125.

Guenon, Rene. "La Tetraktys et le carre de quatre," Etudes Traditionnelles 42 (May 1937).

Guenon, Rene. Symboles fondamentaux de la Science sacree. Paris, Gallimard, 1962 (includes the above article).

Heath, Thomas. A History of Greek Mathematics. 2 vols. Oxford University Press, 1921.

Heath, T. L. The Thirteen Books of Euclid's Elements. 3 vols. Cambridge University Press, 1926.

Heidel, W.A. "The Pythagoreans and Greek Mathematics," American Journal of Philology 61 (1940), 1-33.

Hippocrates. De Hebdomad. Classical Quarterly 65 (1971), 365-88.

Hopper, V.F. Medieval Number Symbolism: Its Sources, Meaning and Influence on Thought and Symbolism. New York, Columbia University Press, 1938.

Junge, Gustave. "Die pythagoreische Zahlenlehre," Deutsche Mathematik 5 (1940), 341-57.

Junge, Gustave. "Von Hippasos bis Philolaus: das Irrationale und die geometrischen Grundbegriffe," Classica et Medievalia 19 (1958), 41-72.

Kleinhammes, Otto. Die Quadratur des Kreises aus dem Geiste der Musik. Wanger im Allgau, J. Kleinhammes, 1949.

Kozminsky, Isidore. Numbers, Their Meaning and Magic. New York, Putnam's Sons, 1927.

Kucharski, Paul. Etude sur la doctrine pythagoricienne de la tetrad. Paris, Les Belles Lettres, 1952.

Michel, P. De Pythagore a Euclide. Paris, Les Belles Lettres, 1950.

Neugebauer. O. The Exact Sciences in Antiquity. Princeton University Press, 1952.

Nicomachus of Gerasa. Introduction to Arithmetic. Trans. by Martin Luther D'Ooge. New York, MacMillan, 1926.

Oliver, G. The Pythagorean Triangle. (1875) San Diego, Wizards Bookshelf, 1975.

Pacioli, Luca. Summa de arithmetica, geometria, proportioni & proportionalita. Venice, 1494.

Pacioli, Luca. De divina proportione. Milan, 1956; and Paris, Librairie du Compagnonnage, 1984.

Robbins, Frank E. "The Tradition of Greek Arithmology," Classical Philology 16 (1921), 97-123.

Saint-Martin, Louis-Claude de. Les Nombres. Paris, Documents martinistes, 1983.

Schnitzler, Gunter, ed. Musik und Zahl: Interdiziplinare Beitrage zum Grenzbereich zwischen Musik und Mathematik. Bonn, Verlag fur systematische Musikwissenschaft, 1976.

Speusippus. Fragments from his work "On Pythagorean Numbers" in Thomas, Greek Mathematical Works, vol. 1, Harvard University Press, 1939.

Taylor, Thomas. The Theoretic Arithmetic of the Pythagoreans. (1816) New York, Samuel Weiser, 1978.

Theon of Smyrna. Mathematics Useful for Understanding Plato. Trans. by R. & D. Lawlor. San Diego, Wizards Bookshelf, 1979.

Thomas, Ivor. Greek Mathematical Works. Vol. I: From Thales to Euclid. Harvard University Press, 1951.

Waerden, B.L. van der. "Die Arithmetik der Pythagoreer," Mathematische Annalen, 1 (1948), 127-53; 2 (1948), 676-700.

Waerden, B.L. van der. Science Awakening: Egyptian, Babylonian and Greek Mathematics. New York, Science Editions, 1961.

Waerden, B.L. van der. Geometry and Algebra in Ancient Civilizations. New York, Springer-Verlag, 1983.

Wasserstein, A. "Theatetus and the Theory of Numbers," Classical Quarterly 52 (1958), 165-79.

Wedberg, A. Plato's Philosophy of Mathematics. Stockholm, Almquist and Wiksell, 1955.

Weiss, C.H. Betrachtung des Dimensionsverhaltnisse in den Hauptkorpem des spharoidischen Systems und ihren Gegenkorpen, im Vergleich mitden harmonischen Verhaltnissen der Tone. Berlin, 1820.

Wescott, Wynn. Numbers: Their Occult Power and Mystic Virtues. (1902) London, Theosophical Publishing House, 1974.

MUSIC

These writings deal with Pythagorean musical and harmonic theory.

Abert, Hermann. Die Musikanschauung des Mittelalters und ihre Grundlagen. Halle, 1905, reprinted Tutzing, Schneider, 1964.

Amy-Sage, Fidele. La Musique et l'esprit. Paris, Voile d'Isis, 1920.

Aristeides Quintilianus. On Music Trans. by T. Mathiesen. Yale University Press, 1983.

Bailly, Edmond. Le Chant des Voyelles comme Invocation aux Dieux Planetaires. (1911) Nice, Belisae, 1976.

Barbour, J. Murray. "The Persistence of the Pythagorean Tuning System," Scripta Mathematica 1 (1932-33), 286-304.

Becker, Oskar. "Fruhgriechische Mathematik und Musiklehre," Archiv fur Musikwissenschaft 14 (1957), 156-164.

Boethius. The Principles of Music. Trans. by Calvin M. Bower. Diss., George Peabody College for Teachers, 1966.

Bonnaire, M.U. De l'influence de la musique sur les moeurs. Vienne, 1856.

Bower, Calvin. "Boethius and Nicomachus, an Essay Concerning the Sources of De Institutione Musica," Vivarium 16 (1978), 1-45.

Cazden, Norman. "Pythagoras and Aristoxenus Reconciled," Journal of the American Musicological Society 11.2-3 (1958), 97-105.

Chailley, Jacques. Nombres et symboles dans le langage de la musique. Paris, Academie
des Beaux-Arts, 1982.

Chamberlain, David S. "Philosophy of Music in the Consolatio of Boethius," Speculum 45 (1970), 80-97.

Cornford, F.M. "The Harmony of the Spheres" in his The Unwritten Philosophy and Other Essays. Cambridge University Press, 1950.

Coste, Charles. L'Influence de la musique. Poligny, 1863.

Crocker, Richard L. "Pythagorean Mathematics and Music," Journal of Aesthetics and Art Criticism, 22.2 (1963), 189-98; 22.3 (1964), 325-36.

Dalberg, J.F.H. von. Untersuchungen uber die Ursprung der Harmonie. Erfurt, 1800.

De Vismes du Valgay, A.-P.-J. Pasilogie. Paris, 1806.

Denereaz, Alexandre. Cours d'harmonie. Paris, Gauthier-Villars, 1906.

During, I. Ptolemaios und Porphyrios uber die Musik. Goteborg, 1934. (Goteborgs Hogskolas Arsskrift 40)

Fabre d'Olivet, Antoine. La Musique expliquee comme Science et comme Art. (1842-50) Paris, Dorbon Aine, 1928.

Gandillot, Maurice. Essai sur la Gamme. Paris, Cavel, 1937.

Godwin, Joscelyn. Cosmic Music: Three Musical Keys to the Interpretation of Reality. West Stockbridge, Mass., Lindisfarne Press, 1987.

Godwin, Joscelyn. Music, Mysticism and Magic: A Sourcebook. London, Routledge & Kegan Paul, 1986.

Godwin, Joscelyn. Harmonies of Heaven and Earth. London, Thames & Hudson, 1987.

Godwin, Joscelyn. "The Golden Chain of Orpheus: A Survey of Musical Esotericism in the West," Temenos 4 (1985), 7-25; 5 (1985), 211-39.

Henderson, Isobel. "Ancient Greek Music," New Oxford History of Music, Oxford University Press, 1957, 336-403.

Henry, Charles. "Conception psycho-physique de la Gamme," Bull. de l'Inst. Gen. Psychologique 21.1-3 (1921).

Ikhwan al-Safa. Epistle on Music. Trans. by A. Shiloah. Tel-Aviv University, Department of Musicology, 1978.

Junge, G. "Die Spharenharmonie und die pythagorisch-platonische Zahlenlehre," Classica et Medievalia 8 (1947), 183-94.

La Borde, J.B. de. Memoire sur les Proportions Musicales, le genre enharmonique des grecs, et celui des modernes. Paris, 1781.

Le Voile d'Isis, special number on "La Musique dans ses rapports avec l'Esoterisme," Apr. 1928.

Levarie, Siegmund and Levy, Ernst. Musical Morphology. Kent State University Press, 1983. Levarie, Siegmund, and Levy, Ernst. Tone: A Study in Musical Acoustics. Kent State University Press, 1968.

Levin, Flora R. The Harmonics of Nicomachus and the Pythagorean Tradition. (American Classical Studies, No. 1.) New York, Interbook, 1975.

Liebard, Louis. Anachronie Musicale, ou la pyramide inversee. Besancon, Resonances, 1979.

Lohmann, Johannes. "Die griechische Musik als mathematische Form," Archiv fur Musikwissenschaft 14 (1957), 147-155.

Lucas, Louis. Une Revolution dans la Musique: essai d'application, a la musique, d'une theorie philosophique. Paris, 1849.

McClain, Ernest G. "Plato's Musical Cosmology," Main Currents in Modern Thought 30.1 (1973), 34-42.

McClain, Ernest G. "Musical Marriages in Plato's Republic," Journal of Music Theory 18.2 (1974), 242-72.

McClain, Ernest G. "A New Look at Plato's Timaeus," Music and Man 1.4 (1975),
341-60.

McClain, Ernest G. The Myth of Invariance. New York, Nicolas-Hays, 1976.

McClain, Ernest G. The Pythagorean Plato: Prelude to the Song Itself. York Beach, Nicolas-Hays, 1984.

Meyer-Baer, Kathi. Music of the Spheres and the Dance of Death. Princeton University Press, 1970.

Millet, Yves. "La primaute de la gamme dite de Pythagore: son symbolisme cosmique," Etudes Traditionnelles 59 (1958), 37-44, 138-161.

Montargis, Frederic. De Platone Musico. (1886) Utrecht, Joamchimsthal, 1976.

Mountford, J.F. "The Musical Scales of Plato's Republic," Classical Quarterly 17 (1923), 125-136.

Paul, Maela and Muxelhaus, Patrick. Le Chant Sacre des Energies. Paris, Presence, 1983.

Pfrogner, Hermann. Lebendige Tonwelt: zur Phanomen Musik. Munich, Langen Muller, 1976.

Portnoy, Julius. Music in the Life of Man. New York, Holt, Rinehart and Winston, 1963.

Provost, Prudent. La Musique renovee selon la synthese acoustique. Paris, Societe francaise d'editions litteraires et techniques, 1931.

Reinach, Theodore. "La Musique des Spheres," Revue de Etudes grecques 13 (1900), 432-49.

Rice, Isaac. What is Music? New York, 1875.

Roussier, P.J. Memoire sur la Musique des Anciens. (1770) New York, Broude. 1966.

Rudhyar, Dane. The Magic of Tone and the Art of Music. Boulder, Shambhala, 1982.

Scott, Cyril. Music: Its Secret Influence throughout the Ages. (1933) New York, Weiser, 1969.

Thamar, Jean. "Notion de la Musique Traditionnelle," Etudes Traditionnelles 48 (1947), 281-293, 336-349; 49 (1948), 58-67, 106-112, 240-253, 301-308, 348-356; 50 (1949), 216-227, 303-318.

Troupenas, E. "Essai sur la Theorie de la Musique deduite du principe metaphysique," Revue musicale 17 (1832), 129-31.

Vassiliadou, Maria. "Le pythagorisme et la musique," in Musique et Philosophie, Colloque de Dijon, 1983, 15-26.

Villoteau, G.A. Memoire sur la possibilite et l'utilite d'une theorie exacte des principes
narurels de la Musique. Paris, 1807.

Waerden. B.L. van der. "Die Harmonielehre der Pythagoreer," Hermes 78 (1943), 163-199.

Walker, D.P. "Kepler's Celestial Music," Journal of the Warburg and Courtauld Institutes 30 (1967), 228-50.

ASTRONOMY

These writings deal with Pythagorean astronomy and cosmology.

Azbel (= Emile Chizat). Harmonie des mondes: Loi des distances et des harmonies planetaires. Paris, Hugues-Roben, 1903.

Burch, G.B. "The Counter-Earth," Osiris (1954), 267-94.

Dicks, D.R. Early Greek Astronomy to Aristotle. Ithaca, Cornell University Press, 1970.

Dreyer, John Louis Emil. History of the Planetary Systems from Thales to Kepler. Cambridge University Press, 1906.

Guenon, Rene. "Le symbolisme du Zodiaque chez les pythagoriciens," Etudes Traditionnelles 43 (June 1938).

Guenon, Rene. Symboles fondamentaux de la Science sacree. Paris, Gallimard, 1962 (includes the above article).

Heath, Thomas L. Greek Astronomy. New York, AMS Press, 1969.

Waerden, B.L. van der. Die Astronomie der Pythagoreer. Amsterdam, Royal Academy, 1951.

MEDIEVAL AND RENAISSANCE

These studies deal primarily with Pythagorean thought in the Medieval and Renaissance periods.

Africa, Thomas W. "Copernicus' Relation to Aristarchus and Pythagoras," Isis 52 (1961), 403-409.

Amman, P.J. "The Musical Theory and Philosophy of Roben Fludd," Journal of the Warburg and Courtauld Institutes, 30 (1967), 198-227.

Boethius. De arithmetica. Edited with commentary by Girardus Rufus. Paris, 1521.

Boethius. De musica. Oscar Paul, ed. Leipzig, 1872.

Bongo, Pietro. Myticae numerorum significationis liber. Bergamo, 1585. Expanded ed., Basel, 1618, entitled De Numerorum Mysteria.

Clulee, Nicholas. The Glass of Creation: Renaissance Mathematics and Natural Philosophy. Diss., University of Chicago, 1973.

Debus, Allen G. "Mathematics and Nature in the Chemical Texts of the Renaissance," Ambix 15 (1968), 1-28.

Fludd, Robert. Utriusque cosmi historia scilicet et minoris metaphysica, physica atque technica historia. Oppenheim, 1621.

Giorgio, Francesco. De harmonia mundi. Venice, 1525.

Godwin, Joscelyn. Roben Fludd: Hermetic Philosopher and Surveyor of Two Worlds. London, Thames & Hudson, 1979.

Hart, Thomas Elwood. "Calculated Casualties in Beowulf: Geometrical Scaffolding and Verbal Symbol," Studia neophilologica 53 (1981), 3-35.

Hart, Thomas Elwood. "Twelfth-Century Platonism and the Geometry of Textual Space in Hartman's Iwein: A 'Pythagorean' Theory," Res publica litterarum 2 (1979), 81-107.

Heninger, S.K., Jr. "Some Renaissance Versions of the Pythagorean Tetrad," Studies in the Renaissance 8 (1961), 7-35.

Heninger, S.K., Jr. "Pythagorean Cosmology and the Triumph of Heliocentrism," in Le soleil a la renaissance. Presses universitaires de Bruxelles, 1965, 33-53.

Heninger, S.K., Jr. Touches of Sweet Harmony: Pythagorean Cosmology and Renaissance Poetics. San Marino, CA, Huntington Library, 1974.

Heninger, S.K., Jr. The Cosmographical Glass: Renaissance Diagrams oft he Universe. San Marino, CA, Huntington Library, 1977.

Hopper, V.F. Medieval Number Symbolism: Its Sources, Meaning and Influence on Thought and Symbolism. New York, Columbia University Press, 1938.

Ingpen, William. The Secrets of Numbers. London, 1624.

Kayser, Hans. Ein harmonikaler Teilungskanon. Zurich, Occident, 1946.

Kepler, Johannes. Mysterium cosmographicum. Tubingen, 1596.

Kepler, Johannes. Harmonice mundi. Linz, 1619.

Kircher, Athanasius. Musurgia universalis sive ars magna consoni et dissoni in x libros digesta. Rome, 1650.

Kircher, Athanasisus. Arithmologia: sive de abditis numerorum mysteriis. Rome, 1665.

Koyre, A. Metaphysics and Measurement: Essays in the Scientific Revolution. Harvard University Press, 1968.

Macrobius. Commentary on the Dream of Scipio. Trans. and ed. W. H. Slahl. New York, Columbia University Press, 1952.

Martianus Capella. De Nuptiis Philologiae et Mercurrii. Ed., A. Dick. Leipzig, 1925.

Munxelhaus, Barbara. Pythagoras Musicus: zur Rezeption der pythagorischen Musiktheorie als quadrivialer Wissenschaft im Lateinischen Mittelalter. Bonn, Verlag fur systematische Musikwissenschaft, 1976.

Nasr, Seyyed Hossein. An Introduction to Islamic Cosmological Doctrines. (Rev. ed.) Boulder, Shambhala, 1978.

Patrides, C.A. "The Numerological Approach to Cosmic Order During the English Renaissance," Isis 49 (1958), 391-397.

Reuchlin, Johann. On the Art of the Kabbalah. (1516) Trans. by Martin & Sarah Goodman. New York, Abaris Books, 1983.

Rosen, Edward. "Was Copernicus a Pythagorean?" Isis 53 (1962), 504-509.

Walker, D.P. Studies in Musical Science in the Late Renaissance. London, Warburg Institute, 1978.

Yates, Francis A. The Occult Philosophy in the Elizabethan Age. London, Routledge & Kegan Paul, 1979.

SACRED GEOMETRY

The writings in this section are concerned with Pythagorean and Platonic geometry in a theoretic and practical sense. Also included are works dealing with the application of "Pythagorean" canons of proportion in sacred architecture.

Bond, F.B., and Lea, T.S. Gematria: A Preliminary Investigation of the Cabala. (1917) London, RILKO, 1977.

Bond, F.B., and Lea, T.S. The Apostolic Gnosis. London, RILKO, 1979. (On Greek gematria.)

Bouleau, Charles. The Painter's Secret Geometry. Trans. by J. Griffin. Paris, 1963.

Brunes, Tons. Secrets of Ancient Geometry. 2 vols. Copenhagen, Rhodos, 1967.

Caskey, L.D. Geometry of the Greek Vases. Boston, 1922.

Colman, Samuel. Nature's Harmonic Unity. (1912) New York, Blom, 1971.

Critchlow, Keith. "The Platonic Tradition on the Nature of Proportion," in Lindisfarne Letter 10.

Critchlow, Keith. Islamic Patterns: An Analytical and Cosmological Approach. New York, Schocken Books, 1976.

Critchlow, Keith. Order in Space: A Design Sourcebook. New York, Viking, 1965.

Critchlow, Keith. Time Stands Still: New Light on Megalithic Science. New York, St. Martins Press, 1982.

Doczi, Gyorgy. The Power of Limits: Proportional Harmonies in Nature, An and Architecture. Boulder, Shambhala, 1981.

Ghyka, Matila C. Le Nombre d'Or. 2 vols. Paris, Gallimard, 1931.

Ghyka, Matila C. The Geometry of An and Life. New York, Dover, 1977.

Gougy, Charles. L 'Harmonie des proportions et des formes dans l'architecture d 'apres les lois de l'harmonie des sons. Paris, Charles Massin, 1925.

Guenon, Rene. The Symbolism of the Cross. London, Luzac, 1975.

Hambridge, Jay. Dynamic Symmetry: The Greek Vase. 1920.

Hambridge, Jay. The Parthenon and other Greek Temples: Their Dynamic Symmetry. 1924.

Hambridge, Jay. The Elements of Dynamic Symmetry. Yale University Press, 1948.

Hudson, H.P., ed. Ruler and Compass. Longmans Green, 1915.

Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. New York, Dover, 1970.

Iverson, E. Canon and Proportions in Egyptian Art. Warminster, Aris and Phillips, 1975.

Kielland, E.C. Geometry in Egyptian Art. London, Tiranti, 1955.

Lawlor, Robert. "Ancient Temple Architecture," in Lindisfarne Letter 10.

Lawlor, Robert. Sacred Geometry: Philosophy and Practice. London, Thames and Hudson,
1982.

Lesser, G. Gothic Cathedrals and Sacred Geometry. 3 vols. London, Tiranti, 1957.

Lindisfarne Letter No. 10, "Geometry and Architecture." West Stockbridge, MA, Lindisfarne Press, 1980.

Lund, F.M. Ad Quatratum: A Study of the Geometrical Basis of Classical and Mediaevel architecture. London, Batsford, 1921.

Macaulay, Anne. "Apollo: The Pythagorean Definition of God," in Lindisfarne Letter 14.

Michell, John. City of Revelation. London, Sphere, 1978. (On Greek gematria.)

Purce, Jill. The Mystic Spiral: Journey of the Soul. London, Thames & Hudson, 1974.

Saint-Yves d'Alveydre. L'Archeometre. (1912) Paris, Dorbon Aine, 1979.

Scholfield, P,H. The Theory of Proportion in Architecture. Cambridge, 1958.

Schwaller de Lubicz, R.A. The Temple in Man. Trans. by R. & D. Lawlor. Brookline, MA, Autumn Press, 1977.

Simson, Otto von. The Gothic Cathedral. New York, Pantheon, 1956.

Stirling, William. The Canon: An Exposition of the Pagan Mystery Perpetuated in the Cabala as the Rule of All the Am. (1897) London, RILKO, 1981 (On Greek gematria.)

Sunderland, Elizabeth R. "Symbolic Numbers and Romanesque Church Plans," Journal of the Society of Architectural Historians 18 (1959), 94-103.

Talemarianus, Petrus. De L'Architecture Naturelle. Paris, Vega, 1950.

Thompson, D.W. On Growth and Form. Cambridge University Press, 1917.

Williams, R. The Geometrical Foundation of Natural Structure. New York, Dover, 1979.

WHOLE SYSTEMS

Included here are works pertaining to the application of the Pythagorean approach. especially the principles of harmonics, to scientific, artistic or philosophical synthesis.

Amoux, George. Musique Platonicienne, Arne du Monde. Paris, Dervy, 1960.

Azbel (=Emil Chizat). Le Beau et sa Loi: Loi de l'Action et des Nombres, Loi de l'Harmonie, Loi de l'Intelligence. Paris, Hugues-Robert, 1899.

Bailly, Edmond. Le Son dans la Nature. Paris, L'Art Independant, 1900.

Bindel, Ernst. Die Zahlengrundlagen der Musik im Wandel der Zeiten. 2 vols. Stuttgart, Freies Geistesleben, 1950, 1951.

Blair, Lawrence. Rhythms of Vision. New York, Schocken, 1975.

Briseux, C.E. Traite du Beau essential dans les Arts. Paris, 1752.

Britt, Ernest. "La Synthese de la Musique," Annales du XXe Siecle, 2 (1914), reprinted Paris, Vega, 1938.

Britt, Ernest. Gamme Siderale et Gamme Musicale: Etude Paleosophique. Paris, Aux Ecoutes, 1924. Britt, Ernest. La Lyre d'Apollon. Paris, Vega, 1931.

Choisnard, Paul. La Chaine des Harmonies et la Spirale dans La Nature. (2nd augmented ed.) Paris, Ernest Leroux, 1926.

Denereaz, Alexandre. La Gamme, ce probleme cosmique. Zurich, Hug, 1939.

"Francisque," Le Secret de Pythagore devoile, ou le fond de La musique. Rochefort, 1869.

Franz, Marie-Louise von. Number and Time. London, Rider, 1974.

Fuller, Buckminster. Synergetics. New York, Macmillan, 1975.

Fuller, Buckminster and Applewhite, Edgar. Synergetics II. New York, Macmillan, 1979.

Griveau, Maurice. Programme d'une science idealiste. Paris, Revue Moderne d'Esthetique, 1896.

Griveau, Maurice. La sphere de la beaute Paris, Alcan, 1901.

Guyot, E. La Boussole de l'harmonie universelle, esthetique applicable aux arts des sons, de la couleur et de La forme. Albi, Corbiere et Julien, 1894.

Hill, Clarence. Harmonia Harmonica. 3 vols. Bournemouth, Author, 1920-1935.

Kayser, Hans. Akroasis: The Theory of World Harmonics. Trans. by R. Lilienfeld. Boston, Plowshare, 1970.

Kayser, Hans. Orphikon: eine harmonikale Symbolik. Basel, Schwabe, 1973.

Kayser, Hans. Lehrbuch der Harmonik. Zurich, Occident, 1950.

Labat, J.B. Les Nombres appliques a la Science Musicale. Bordeaux, 1861.

Lacuria, P.F.G. Les Harmonies de l'Etre exprimees par les Nombres. 2 vols. Paris, 1847.

Landsberg, G.F. "Essai Tentative d'adaptation des Lois Musicales a une clef des Lois Universelles," Voile d'Isis, 1914, 385-394.

Lange, Anny von. Mensch, Musik und Kosmos. 2 vols. Freiburg, Die Kommenden, 1960.

Lawlor, Robert. "Pythagorean Number as Form, Color and Light," in Lindisfarne Letter 14.

Levarie, Sigmund and Levy, Ernst. "The Pythagorean Table," Main Currents in Modern Thought 30.4 (1973), 117-29.

Levy, Ernst. "The Pythagorean Concept of Measure," Main Currents in Modern Thought 21.3, (1965), 51-57.

Malfatti de Montereggio, Johann. Anarchie und Hierarchie des Wissen. Leipzig, 1845.

Malfatti de Montereggio, Johann. Etudes sur la mathese (= French trans. of the above). Paris, A. Frank, 1849.

Neroman, Dom. La Lecon de Platon. Paris, Arma Artis, 1983.

Pouyaud, Robert. "Astrologie et Harmonie coloree," L'Atelier de la Rose, 1951, 36-40.

Schmidt, Thomas Michael. Musik und Kosmos als Schozpfungswunder. Frankfurt, Schmidt, 1974.

Spitzer, L. Classical and Christian Ideas of World Harmony. Johns Hopkins Press, 1963.

Swiecianowski, Jules. L'echelle musicale comme loi de l'harmonie dans l'universe et dans l'art. Warsaw, 1881.

Thimus, Albert von. Die harmonikale Symbolik des Alterthums. 2 vols. (1868, 1876) Hildesheim, Olms, 1972.

Wagner, Johann Jacob. Von der Natur der Dinge. Leipzig, 1803.

Zajonc, Arthur G. "The Two Lights," in Lindisfarne Letter 14.

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Figure 20. MAP OF THE ANCIENT WORLD

INDEX OF PROPER NAMES

Abaris the Hyperborean, 80-81, 109-10, 121;
called air-walker, 91, 129
Abroteles, 121
Abrotelia, 122
Achmonidas, 121
Acusiladas, 121
Adicus, 121
Aeetius, 121
Aegon, 121
Aemon, 121
Aeneas, 121
Aethalides, 132, 142
Ageas, 121
Agelas, 121
Agesarchus, 121
Agesidamus, 121
Agylus, 121
Alcaeus, 99
Alceas, 121
Alcimachus, 119, 121
Alcimadas, 156
Alcmaeon, 28, 83, 121
Alexander, 148
Aliochus, 121
Alopecus, 121
Amaerus, 121
Amasis, 124, 141
Ammonius Saccas, 41
Anaxagoras, 156
Anaximander, 59, 123
Anaximenes, 155
Ancaeus, 58, 123
Androcles, 125
Androcydes, 93
Andron, 49
Antimedon, 121
Antimenes, 121
Anthen, 121
Antiphon, 124, 141
Apollodorus, 144
Apollonius, 40, 117, 123
Arceas, 121
Archemachus, 121
Archippus, 117, 121, 134-35, 152
Archytas, 37-38, 83, 105, 117, 120, 121;
fragments, 177-201
Aresas, 120
Arignota, 124
Arimnestus, 123
Aristaeus, 83, 120
Aristangelus, 121
Aristeas, 91, 121
Aristides, 121
Aristippus, 121, 147
Aristocleia, 131
Aristoclidas, 121
Aristocrates, 89, 100, 121
Aristomenes, 121
Aristotle, 49, 63, 145-46, 149; interest in the
Pythagorean school; on Limit and Unlimited, 23; writings on the Pythagoreans
Aristoxenus, 38-39, 49, 113, 117, 125, 127,
135, 141, 145-46, 154; fragments, 243
Arytus, 121
Asteas, 121
Aston, 143
Astraeus, 125
Astylus, 121
Athamas, 121
Athosion, 121
Augustine, 42
Austophon, 151
Auticlides, 144
Autocharidas, 121

Babelyma, 122
Bathylaus, 121
Bias, 59
Bitale, 93
Boethius, 43
Brontinus, 121, 152
Bryas, 121
Bryo, 122
Brysson, 83
Bulagoras, 120
Butherus, 121
Buthius, 121
Byndacis, 122

Calais, 121
Callibrotus, 121
Callicratidas, fragments, 235-37
Carophantidas, 121
Cebes, 38
Cerambus, 121
Charondas, 83, 89, 100, 122. 145; fragments,
231-33
Chilas, 121
Chilonis, 122
Chrysippus, 122
Cicero, 40
Cleaechmas, 122
Cleanor, 121
C1earatus, 121
Clement, 42
Cleon, 121
Cleophron, 121
Cleosthenes, 121
Clinagoras, 121
Clinias, 105, 114, 120-21; fragments, 265
Cornford, F.M., 22, 48
Cranous, 121
Cratesiclea, 122
Cratinus, 151
Crito, 121; fragments, 251-52
Cronius, 41
Cylon, 37, 75, 116, 119, 134

Dacidas, 121
Damarmenus, 121
Damocles, 121
Damo, 93, 152
Damon, 113, 121, 135
Damotages, 121
Dardaneus, 121
Deanax, 121
Delatte, Armand, 163
Democedes, 119
Democritus, 123
Demon, 121
Demosthenes. 121
Dexithes, 121
Dicaerchus, 49, 121, 126, 152
Dicas, 121
Dicon, 121
Dillon, 40
Dimachus, 119
Dinarchus, 120-21
Dinocrates, 121
Diocles, 117, 154, 121
Diodorus, 119-20
Diogenes. 125, 129
Diogenes Laertius, 141
Dion, 105
Dionysius, 103-104, 113, 135
Dionysophanes, 125
Diotogenes, fragments, 221-24
d'Olivet, Fabre, 13
Drymon, 121
Duris, 123
Dymas, 121

Echecrates, 38, 117, 121, 154
Echecratia, 122
Ecphantus, 121; fragments, 257-259
Eiriscus, 121
E1ecaon, 89
Empedocles, 83, 121, 152, 155-56; called wind-stiller, 91, 129
Empedus, 121
Epaminondas, 117, 134, 143
Epichannus, 120
Epimenides, 83, 110, 142; called expiator, 91, 129
Epiphron, 121
Episylus, 121
Eratosthenes, 154
Eratas, 121
Eubulus, 89
Eudorus, 40
Eudoxus, 124
Euelthon, 121
Eunomus, 141
Eunostus, 123, 125
Euphemus, 121
Euphorbus, 71, 132, 142
Eurycrates, 121
Eurymedon , 121
Eurymenes, 103•104, 126
Euryphamus, 102-03; fragments, 245-46
Euryphemus, 121
Eurytus, 83, 91, 94, 120, 121, 154
Euthycles, 121
Euthynus, 121
Evaeus, 121
Evandrus, 121
Evanor, 121
Evetes, 121

Favorinus, 144, 154
Ficino, Marsilio, 13, 42

Gartydas, 120
Glorippus, 121
Guthrie, Kenneth Sylvan, 13, 17
Glycinus, 121
Gyptius, 121

Hall, Manly P., 13
Helicaon, 100, 121
Heloris, 121
Heracleides, 33, 142-43, 152-53
Heraclitus, 142, 142n
Hermippus, 141, 144, 152, 156
Hermodamas, 123, 126
Hermotimus, 132, 142
Hestiaeus, 121
Hiero, 120
Hierocles, ethical fragments, 275-86
Hieronymus, 147
Hipparchides, 121
Hipparchus, 75, 152; fragments, 247-48
Hippasus, 83, 119, 121, 143
Hippobotus, 103, 135, 152
Hippodamus, fragments, 215-20
Hippodomas, 77
Hippomedon, 79, 121
Hippon, 121
Hipposthenes, 121
Hippostratus, 121

Iambkichus, 42, 271, 321
Icmus, 121
Itmaeus, 121

Justin Martyr, 42

Lacon, 121
Lacrates, 121
Lacritus, 121
Laphion, 121
Lasthenia, 122
Leocritus, 121
Leocydes, 121
Leon, 121
Leophron, 121
Leptines, 121
Leucippus, 83
Levy, Ernst, 48
Litagus, 120
Lycon, 121
Lycus, 124
Lyramnus, 122
Lysiades, 122
Lysibius, 121
Lysis, 38, 75, 83, 102-03, 121, 134-35, 143,
152

Macrobius, 42
Malion 121
Marmacus, 141
Martianus Capella, 42-43
Medici, Cosimo de', 42
Megistias, 121
Melanippus, 121
Melisias, 121
Melissus, 121
Menestor, 121
Menon, 121
Meton, 119, 121
Metopus, 121; fragments, 249
Milo, 83, 117, 134, 151-52
Milon, 121
Miltiades, 89, 121
Mimnomachus, 121
Mnesarchus (father of Pythagoras), 58-59,
123, 125, 141; (son of Pythagoras), 120,
137
Mnesibulus, 121
Mnesimachus, 151
Moderatus, 41
Moses, 60
Mya, 122
Myes, 121
Myia, 124, 137
Myllia, 135
Myllias, 103-04

Nastas, 121
Nausitheus, 122
Nausithus, 89
Neanthes, 103, 123, 135, 155
Neocritus, 122
Nicomachus, 41, 117, 126, 135
Nigidius Figulus, 40
Ninon, 119
Nisleadusa, 122
Numenius, 41

Ocellus. 121; On the Nature of the Universe.
203; On Laws, 213
Occillus, 121
Odius, 121
Onatus, 121
Opsimus, 121
Oresandrus, 121
Orestadas, 121

Paction, 121
Parmenides, 121, 145, 155-6,
Parmiseus, 12 I
Parthenis, 58
Pempelus, fragments, 261
Perialus, 75
Perictyone, fragments, 239-41
Phaedon, 121
Phaenec1es, 121
Phalaris, 109-10
Phanto, 117
Phanton, 121, 154
Pherecydes, 59, 102, 116-7, 123, 126, 134,
141, 152
Philo, 40
Philodamus, 121
Philolaus, 38, 83, 94, 105, 120-1, 154;
fragments, 167-75; music heard near his
tomb, 91
Philonidas, 121
Philostratus, 40
Philtis, 122
Phintias, 113, 121, 135
Photius, 137
Phrontidas, 121
Phrynichus, 121
Phyntis, fragments, 263
Phytius, 89, 100, 121
Pisicrates, 121
Pisirrhonde, 122
Pisyrrhydus, 121
Plato, 38, 52, 105, 299; letter to Archytas,
177-8
Plotinus, 42
Plutarch, 40-1
Polemaeus, 121
Polemarchus, 121
Poliades, 121
Polus, fragments, 235

Rhexibius. 121
Rhodippus, 121

Sara, 137
Sextus, Pythagorean Sentences, 267-71
Sillus, 121
Simmias, 38
Simus, 121, 124
Smichias, 121
Sosicrates, 143
Sosistratus, 121
Sosthenes, 121
Sostratius, 121
Speusippus, 38
Spintharus, 105
Sthenidas, fragments, 255
Sthenonidas, 121
Syllus, 94

Taylor, Thomas, 13
Te1auges, 93, 124, 137, 152-3
Thales, 59, 125
Theaetetus, 100, 154
Theages, 119; fragments, 225-8
Theano, 90, 120, 122, 124, 137, 152-3
Themistoclea, 147
Theoc1es, 89
Theodorus, 121
Theon, 41; on the Tetraktys, 317-9
Theorides, 120
Thesleff, Holger, 39
Thestor, 114
Thompson, Francis, 44
Thraseus, 121
Thrasydamus, 121
Thrasymedes, 121
Thymaridas, 83, 93, 114, 121
Timaeus (of Croton), 121; (of Locri), On the World and the Soul, 287-96; (the Parian), 121; (of Tauromenium), 49, 124, 144, 155
Timares, 89, 121
Timaratus, 100
Timasius, 121
Timesianax, 121
Timon, 150
Timosthenes, 121
Timycha, 103-4, 122, 135
Tyrrhenus, 123, 125, 141
Tyrsenis, 122
Tyrsenus, 121

Whitehead, Alfred North, 19

Xenocades, 121
Xenon, 121
Xenophanes, 150
Xenophantes, 121
Xenophilus, 38, 117, 121, 154
Xentas, 121

Zaleucus, 83, 89, 100, 121, 145; fragments, 229-30
Zalmoxis, 83, 100, 125, 141
Zoilus, 141
Zopyrus, 121
Zoroaster, 20, 125

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INDEX OF SELECT TOPICS

Alcmaeon, 151
analogia, 29, 37
Anaximenes, Letter to, of Pythagoras, 155
Apeiron, see Unlimited
Apollo, 30; as the One, 41; see also Pythios
Apolionius of Tyana, The Life of, 40
Apothems of Aristoxenus, 243
Archytas, The Fragments of, 177-201
Arithmetic, Introduction to, 41
arithmetic mean; see mean, arithmetic
Arithmetic, On, 43
arithmology, 321; in Plutarch, 40
Athletics, On, 62

catharsis, 31, 36, 84-85
Chronicles (of Duris), 123
church fathers, 297-98
Clinias, A Fragment of, 265
Common Mathematical Science, On the, 42
Commentaries (of Favorinus), 144
Constitution of the Delians, 145
counter-earth, 129, 170, 301
Croton, 143

decad, 133, 137, 171, 175
Description of the Earth, 124
Divergence of the Academics from Plato, On
the, 41
dodecahedron, 116
dogmata agrapha, 38
doxographers, 307; fragments from, 307-314
Dream of Scipio, Commentary on the, 42
dualism, 33-34, 51-52
duality, 22
dyad, 21, 133, 137; indefinite, 24, 38, 42,
148, 308

elements, four, 174, 206-208, 291-292
Ethical Fragments of Hierocles, 275-86
Exhortation to Philosophy, 42

Fables, 123
Felicity of Families, On the, 235-37
friendship, 112-114

gematria, 41, 53
geometric mean; see mean, geometric
Geometry, Introduction to, 41
gnomon, 24, 300
Gods, Concerning the, 80, 93
Golden Verses of Pythagoras, The, 163-65
Good, On the, 41
Greek dialects, the antiquity of, 115

harmonic mean; see mean, harmonic
harmonic proportion (6:8::912), 24-28, 169,
as "metaphysical octave, " 50; see also
harmonics, musical proportions
harmonics, 24-28, 86-88
Harmonics, Manual of, 41
harmony, 22; etymology of, 51; and health,
28; and justice, 36; of Limit and
Unlimited, 168; as mediation, 25; as a
blending of opposites, 28; in the soul, 33;
and symmetry, 28
Harmony of a Woman, On the, 239-41
"Hearers;" see Pythagorean school,
akousmatikoi
Helothales, the Father of Epicharmus of Cos,
143
Hermes, 69, 149
Histories (of Lycus), 124
History of Alexander, 144
History of Philosophy in Ten Books, 42
History of Sicily, 49, 155
Hoopoe, 41
Human Life, Concerning, 245-46
Hymn to Number, 51
Hyperboreans, 13; see also Abaris

Illustrious Virtuous Men, On, 124, 141
Incredible Things Beyond Thule, On the, 125
Indestructibility of the Soul, On the, 41

justice, 253; as proportion, 33; see also, harmony,
philosophy, political, logos
Justice, On, 253

Kingdom, Concerning a, 222-224
Kingdom, On a, (of Ecphantus), 257-59
Kingdom, On a, (of Sthenidas), 255
knowledge, as harmonic element, 34
kosmos, 22, 139, 299

Laws, On, 215
Limit, 22-24, 29, 46, 168, 171-72, 182-83,
251, 301. 304
Lives of Satyrus, Epitome of the, 152
logos, 25

Marriage of Philology and Mercury, 42
Mathematics Useful for Understanding Plato,
41
mathematikoi; see Pythagorean school,
mathematikoi
maxims, Pythagorean; see Pythagorean
school, symbols and maxims
mean, arithmetic, 25; geometric, 28; harmonic,
25
microcosm, man as, 31, 139
Middle Platonism, 40-42
Miscellaneous History, 144-45, 154
monad, 40, 42, 133, 137, 148, 171, 179; as
origin of all things, 138
monochord, 24-28
Moralia, 40
Muses, the, 30
music, its effect on the soul, 34; of the spheres, 129
music theory; see harmonics
music therapy, 35, 84-85
Music, On, 43
musical proportion; see harmonic proportion
musical ratios, 168-69, 185, 191, 327; leimma
Or semitone (243:246), 27-28, 50; octave
(1:2), 25, 27; perfect fifth (2:3), 25, 27; perfect fourth (3:4), 25,
27; whole tone, 27, 28

Natural, Ethical and Divine Conceptions
which are Perceived in the Science of Numbers, On the, 42
Natural Philosophers, Account of, 147
Nature, On, (of Philolaus), 167
Nature, On, (of Pythagoras), 80
Nature of the Universe, On the. 203-11
Neoplatonism, 41-42
Neopythagoreanism, 39-42
Nicomachus' Introduction to Arithmetic. Commentary
on, 42
Number, as first principle, 301; as immanent,
34-35; as paradigm, 21, 28-30, 169, 171;
and the soul, 33; as source of divine
natures, 93; the study of, 34; as transcendent,
34-35
numbers, as emanations of the One, 11, 50
Numbers, On, 41

One, the, 11-12, 21, 133, 137, 179, 302-303,
308; above being, 41; and the Good, 38,
41; above monad and dyad, 40
oracle at Delphi, 77
Orphism, 31, 36

Parents, On, 261
Peras; see Limit
Philolaus. The Fragments of, 167-75
philosophy, aesthetic, 52; as the care of the
soul, 32-33; educational, 33-36; ethical,
37, 185-190, 225-228, 249; political,
36-37, 99-103, 190-193, 217-220,
222-224, 255, 257-259; as purification,
see catharsis; term invented by
Pythagoras, 30; as a way of life, 30-33;
of whole systems, 43-48
Physics, 156
Piety, On, 143
Place, On, 41
Platonic solids; see regular solids
poiitikoi; see Pythagorean school, politikoi
praxis, 35-36
Preface to the Laws of Charondas the Catanean.
231-33
Preface to the Laws Of Zaleucus the Locrian,
229-30
Prudence and Prosperity, On. 251-52
pseudepigrapha, 39
Pythagoras, addressed boys of Croton, 68; addressed
men of Croton, 66-68; addressed
by a river, 90, 128, 144; addressed
women of Croton, 69-70; addressed young
men of Croton, 65-66; Anaximenes, Letter
10, 155; as the offspring of Apollo, 58,
123; Athletics, On, 62; his birth, 58-59;
biographical sources of information, 49;
brought down eagle, 71, 127-28; Croton,
143; cured by music, 72; descent to underworld,
147, 152; his diet, 130; dietary suggestions,
84; Education. On, 143; his
family, 152-53; Gods. On the, 80, 93;
received golden dart from Abaris the
Hyperborean, 92; possessed a golden t
high, 90, 128, 144; Golden Verses.
163-65; Helothales. the Father of
Epicharmus of Cos, 143; initiations in
Crete, 126; journey to Egypt, 60-61;
meaning of his name, 147; antiquity of
"miracle stories, " 49; use of music,
84-86, 129-30; heard the music of the
spheres, 72; Na1J4re, On, 80, 143-142; his
use of number, 133; his past lives, 71,
132, 142.312; personal habits, 146; Soul,
On the, 143; spoke against Phalaris,
109-110; called himself a philosopher,
143; Politics, On, 143; predicted number
of fish in net, 65, 128; religious observations,
94-96; Sacred Discourse, 93, 95,
119; Sacred Poem, 143; seen in two places
at once, 90, 128; spoke to an ox, 71, 127;
studies in Egypt and Babylonia, 61; tamed
wild bear, 70, 127; transmigration,
126; Universe, On the, 143; various
Pythagorases, 154; writings of, 142-43;
wrote poems under the name of Orpheus,
19, 143
Pythagoras, The Life of, (by Apollonius), 40
Pythagoras, The Life of, (by Aristoxenus),
113, 135
Pythagoras, The Life of, (by Diogenes),
141-56
Pythagoras. The Life of, (by Iamblichus),
57-122
Pythagoras, The Life of, (preserved by
Photius), 137-40
Pythagoras, The Life of, (by Porphyry),
123-35
Pythagoras. On, (by Hermippus), 144
Pythagorean, The, 151
Pythagorean Numbers. On, 38
Pythagorean school, akousmatikoi, 30-31, 63,
76, 130; cenobites, 63; daily program,
81-82; mathematikoi, 31, 76, 130, 137;
list of members, 121-22; politikoi, 31;
revolt against 37-38, 116-20, 134, 151-52;
the succession, 120; symbols or maxims,
78, 83, 131-32, 146, 159-61; teachings,
see philosophy
Pythagorean Symbols, On, 93
Pythagorean Woman, The, 151
Pythagoreans, On the, (by Aristotle), 49, 38,
63, 149
Pythios (Apollo), 147

Quadrivium, the, 34

regular solids, 34, 174, 291, 309, 315, 315
reincarnation; see transmigration
Reincarnation, On, 41
Republic, On a, 217-20
resonance, 34
Rules of Education, 145

Sacred Discourse, 93. 95, 119
Sanctity, On, 221
Secret Doctrines of Plato, On the, 41
Select Sentences of Sextus the Pythagorean.
267-70
Silli, 150
Sotion, Abridgement of. 143
soul, associated with body through number,
174; divided in three parts, 32-33
Soul, On the, 143
stereometry, 34
"Students;" see Pythagorean school.
mathematikoi
Successions of the Philosophers, (by Alexander),
148
Successions of the Philosophers. (by
Sosicrates), 143
symbols, Pythagorean; see Pythagorean school, symbols or maxims

Table of Opposites, 23, 34, 130, 301 -02
Tarentines, The, 151
temperance, 103-108; see also philosophy,
ethical, and virtues
Tetraktys, 28-30, 307-308, 312, 77; as the
"Harmony of the Sirens, " 77; symbolism
of, 29
Theology of Arithmetic (by Iamblichus), 42
Theology of Arithmetic (by Nicomachus), 41
theoria, 35-36
therapeia, 35
Tranquility, On. 247-48
transmigration, 36; see also Pythagoras, his
past lives
triad, 133, 300
Tripod, The, 49

Unlimited, 22-24, 46, 168, 171-72, 182, 251,
299-301, 304

Virtue, Concerning, 249
virtues, the 225-28. 249
Virtues, On the, 225-28

Wheel of Generation, 36
Wisdom, On, 181
Woman's Temperance. On. 263-64
World and the Soul, On the, 287-96

Y, Pythagorean The, 158

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FIGURES

FRONTISPIECE. PYTHAGORAS (From Stanley's History of Philosophy, 1687)



FIGURE 1. UNlTY, DUALITY AND HARMONY



FIGURE 2. THE PYTHAGOREAN TABLE OF OPPOSITES



FIGURE 3. SQUARE NUMBER



FIGURE 4. OBLONG NUMBER



FIGURE 5. THE MONOCHORD. String, sounding box and moveable bridge.



FIGURE 6. THE HARMONIC NODAL POINTS AND OVERTONE SERIES ON THE MONOCHORD. The above figure illustrates the reciprocal relation which exists between string length and vibrational frequency. By stopping the string at the geometrical nodal points the harmonic overtones may be individually emphasized.



FIGURE 7. THE HARMONIC PROPORTION



FIGURE 8. THE TETRAKTYS.



FIGURE 9. THE THREE LIVES



FIGURE 10.  IAMBLICHUS



FIGURE 11. COIN FROM CROTON



FIGURE 12. THE PYTHAGOREAN Y



FIGURE 13. THE MUSIC OF THE SPHERES



FIGURE 14. THE MUSIC OF THE SPHERES. Shown in this engraving from Renaissance Italy are Apollo, the Muses, the planetary spheres and musical ratios.



FIGURE 15. A TABLE OF TONE NUMBERS. There is evidence that this table of tone numbers is a later addition to the text. The abbreviations I. and ap. represent the two types of semitones, the leimma and the apotome respectively.



FIGURE 16. THE REGULAR SOLIDS. The regular solids, also known as the Platonic solids, were first described by Plato in his Timaeus. Plato identified the dodecahedron with the cosmic sphere (later identified with aither), and the four other solids with the four elements. Each one of the elemental "molecules" is constructed out of the triangular "atoms" shown below. The five regular solids are the only polyhedra that can be constructed out of the same regular polygons. The archetypal ratios and geometries with which they are associated underlie the structure and divisions of three-dimensional space.



DODECAHEDRON: Aither / 12 Sides
TETRAHEDRON: Fire / 4 Sides
CUBE: Earth / 6 Sides
OCTAHEDRON: Air / 8 Sides
ICOSAHEDRON : Water / 20 Sides

FIGURE 17. THE PLATONIC LAMBDA



FIGURE 18. THE DIVINE MONOCHORD. This particular monochord is tuned in the key of G, while the examples on the right and in the introduction use the key of C. The three top notes on this monochord are incorrectly placed.



FIGURE 19. THE RATIOS OF THE PYTHAGOREAN SCALE



FIGURE 20. MAP OF THE ANCIENT WORLD