The Pythagorean Sourcebook and Library

[admin]

TIMAEUS OF LOCRI:

ON THE WORLD AND THE SOUL

TIMAEUS OF LOCRI is the central character of Plato's dialogue dealing with Pythagorean cosmology but it is uncertain whether or not there ever actually was a Timaeus of Locri who was a Pythagorean.

This particular writing may have gone through several stages of composition. It started out as an epitome of the cosmology of the Platonic dialogue. However, certain important details are not adequately treated and, in general, the cosmology is reduced to a series of statements while the underlying explanations and reasonings are omitted. At this stage the writing could well have been a summary of a student, and it seems likely that the attribution of the writing to Timaeus was added at a later date. Also added at a later date was a table of tone numbers which relate to the division of the world soul.

The best translation of this work is that of Thomas Tobin, published by Scholars Press in 1985, which features a good introduction and extensive notes. Tobin argues persuasively that this work may be seen as a Middle Platonic interpretation of the Timaeus; hence it was probably composed in the first century of, or before, the common era.

While Plato's Timaeus constitutes essential reading for anyone interested in Pythagorean cosmology, the writing reproduced here possesses more significance for the history of Middle Platonism than it does for the study of Pythagorean thought.

ON THE WORLD AND THE SOUL

1. Mind, Necessity, Form and Matter

TIMAEUS OF LOCRI said the following:

Of all the things in the universe there are two causes: Mind, of things existing according to reason; and Necessity, of things [existing] by force, according to the power of bodies. The former of these causes is the nature of the good, and is called God, and the principle of things that are best, but what accessory causes follow are referred to Necessity. Regarding the things in the universe, there exist Form, Matter and the Perceptible which is, as it were, an offspring of the two others. Form is unproduced, unmoved, stationary, of the nature of the Same, perceptible by the mind, and a pattern of such things produced as exist by a state of change: that is what Form is said to be.

Matter, however. is a recipient of impressions, is a mother and a nurse, and is procreative of the third kind of being; for receiving upon itself the resemblances of form, and as it were remoulding them, it perfects these productions. He asserted moreover that matter, though eternal, is not unmoved; and though of itself it is formless and shapeless, yet it receives every kind of form; and that which is around bodies is divisible and partakes of the nature of the Different; and that matter is called by the twin names of Place and Space.

These two principles then are opposite to each other, of which Form is analogous to a male power and a father, while matter is analogous to a female power and a mother. The third thing is their offspring. Being three, they are recognizable by three marks: Form, by mind, according to knowledge; Matter by a spurious kind of reasoning, because it cannot be mentally perceived directly, but by analogy; and their production by sensation and opinion.

2. Creation of the World

BEFORE THE HEAVENS, then, there existed through reason Form and Matter, and the God who develops the best. But since the older surpasses the younger, and the ordered surpasses the orderless, the deity, being good -- on seeing that Matter receives Form, and is altered in every way, but without order -- found the necessity of organizing it, altering the undefined to the defined, so that the differences between bodies might be proportionately related, not receiving various alterations at random. He therefore made this world out of the whole of Matter, laying it down as a limit to the nature of being, through its containing in itself the rest of things, being one, only-begotten, perfect, endowed with soul and reason -- for these qualities are superior to the soulless and the irrational -- and of a sphere-like body, for this is more perfect than the rest of forms.

Desirous then of making a very good production he made it a divinity, created and never to be destroyed by any cause other than the God who had put it in order, if indeed he should ever wish to dissolve it. But on the part of the good there is no rushing forward to the destruction of a very beautiful production. Such therefore being the world, it continues without corruption and destruction, being blessed. It is the best of things created, since it has been produced by the best cause, which looked not to patterns made by hand but to Form in the abstract, and to Existence, perceiving by the mind to which the created thing, having been carefully adjusted, has become the most beautiful. It is even perfect in the realm of sense because its pattern, containing in itself all the living things perceived by mind, left out nothing, being the limit of the things perceived by mind, as this world is of those perceived by sense.

Being solid and perceptible by touch and sight, the world has a share of earth and fire, and of the things between them, air and water; and it is composed of all perfect bodies, which are in it as wholes, so that no part might ever be left out, in order that the body of the Universe might be altogether self-sufficient, uninjured by corruption from without or within; for apart from these there is nothing else, and hence the things combined according to the best proportions and with equal powers, neither rule over, nor are ruled by each other in turn, so that some receive an increase, others a decrease, remaining indissolubly united according to the very best proportions.

3. Proportions of the World Combination

WHENEVER THERE ARE any three terms with mutually equal intervals that are proportionate, we then perceive that, after the matter of an extended string, the middle is to the first, as is the third to it, and this holds true inversely and alternately, interchanging places and order, so that it is impossible to arrange them numerically without producing an equivalence of results. Likewise the world's shape and movement are well arranged; the shape is a sphere, self-similar on all sides, able to contain all shapes that are similar, and the movement endlessly exhibits the change dependent on a circle. Now as the sphere is on every side equidistant from the center, it is able to retain its poise whether in movement or at rest, neither losing its poise nor assuming another. Its external appearance being exactly smooth, it needs no mortal organs such as are fitted to and present in all other living beings because of their wants. The world soul's element of divinity radiates out from the center entirely penetrating the whole world, forming a single mixture of divided substance with undivided form; and this mixture of two forces, the Same and the Different, became the origin of motion, which indeed was not accomplished in the easiest way, being extremely difficult.

Now all these proportions are combined harmonically according to numbers, which proportions were scientifically divided according to a scale which reveals the elements and the means of the soul's combination. Now seeing that the earlier is more powerful in power and time than the later, the deity did not rank the soul after the substance of the body, but made it older by taking the first of unities, 384. Knowing this first, we can easily reckon the double (square) and the triple (cube); and all the terms together, with the complements and eights, there must be 36 divisions, and the total amounts to 114,695.

These are the divisions


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.

4. Planetary Revolutions and Time

GOD THE ETERNAL, the chief ruler of the Universe and its creator, is beheld alone by the mind, but we may behold by sight all that is produced in this world in connection with those parts which are heavenly, and, being ethereal, must be divided into kinds: some relating to Sameness, others to Difference. Sameness draws onward all that is within, with the general motion of the entire sphere of the universe from east to west. Difference draws along all self-moved portions from west to east, fortuitously rolled around and along by the superior power of Sameness. [1]

The movement of the Different, being divided in harmonic proportion, assumes the order of seven circles. Nearest to the earth the Moon revolves in a month, while beyond her the Sun completes his revolution in a year. Two planets run co-equal with that of the sun: Mercury and the star of Hera, the latter of which is also called the star of Venus and the Lightbringer because shepherds and common people, generally not skillful in sacred astronomy, confuse the western and eastern risings. The same star may shine in the west when following the sun at a distance great enough to be viewed in spite of solar splendor; and at another time in the east when, as herald of the day, it rises before the sun, leading it. Because of its running together with the sun, Venus is the Lightbringer frequently but not always; for there are planets and stars of any magnitude seen above the horizon before sunrise, herald of the day. But the three other planets, Mars, Jupiter and Saturn have their peculiar velocities and different years, completing their course while making their periods of effulgence, of visibility, of obscuration and eclipse, accurately rising and setting. Moreover, they complete their appearances conspicuously in the east or west according to their position relative to the sun who, during the day, speeds westward, which during the night it reverses, under the influence of Sameness, while its annual revolution is due to its inherent motion. In consequence of these two kinds of motion it roils out a spiral, moving one degree each day, but is whirled around under the sphere of the fixed stars according to each revolution of darkness and day.

Now these revolutions are by men called portions of time, which the deity arranged together with the world. For before the world the stars did not exist, and hence there was neither year, nor periods of season, by which this generated time is measured, and which is the representation of the ungenerated time called eternity. For as this heaven has been produced according to an eternal pattern, the world of ideas, so was our world-time created simultaneously according to the pattern of eternity.

5. The World's Creation by Geometric Figures

THE EARTH, fixed at the center, becomes the hearth of the Gods and the boundary of darkness and day, producing settings and risings according to the occultation produced by the things that form the boundary, just as we improve our sight by making a tube with our closed hand, to exclude refraction. Earth is the oldest body in the heavens. Water was not produced without earth, nor air without moisture, nor could fire continue without moisture and the materials that are inflammable, so that earth is fixed upon its balance at the root and base of all other substances.

Of produced things, the substratum is Matter, while the reason of each shape is abstract Form; of these two the offspring is Earth and Water, Air and Fire.

This is how they were created. Every body is composed of surfaces, whose elements are triangles, of which one is right-angled, and the other has all unequal sides, with the square of the longer side being thrice the size of the lesser, while its least angle is the third of a right angle. The middle one is the double of the least, for it is two-thirds of a right angle. The greatest is a right angle, being one-and-a-half times greater than the middle one, and the triple of the least. Now this unequal sided triangle is half of an equilateral triangle, cut into two equal parts by a line let down from the apex to the base. In each of these triangles there is a right angle; but in the one, the two sides about the right angle are equal, and in the other all the sides are unequal. Now let this be called a scalene triangle; while the other, the half of the square, is the principle of the constitution of Earth. For the square produced from this scalene triangle is composed of four half-squares and from such a square is produced the cube, the most stationary and steady form in every way, having six sides and eight angles. On this account Earth is the heaviest and most difficult elemental body to move, and its substance is inconvertible, because it has no affinity with the other type of triangle. Only Earth has a peculiar element of the square, while the other triangle is the element of the three other substances, Fire, Air and Water. For when the half triangle is put together six times it produces the solid equilateral triangle, the exemplar of the tetrahedron, which has four faces with equal angles, which is the form of Fire, as the easiest to be moved, and composed of the finest particles. After this ranks the octahedron, with eight faces and six angles, being the element of Air; and the third is the icosahedron, with twenty faces and twelve angles, being the element of Water, composed of the most numerous and heaviest particles.

These then, as being composed of the same element, are transmuted into one another. But the deity made the dodecahedron the image of the Universe, as being the nearest to the sphere. Fire then, by the fineness of its particles, passes through all things; and Air through the rest of things, with the exception of Fire; and Water through the Earth. All things are therefore full, and have no vacuum. They cohere by the revolving movement of the Universe, and are pressed against and rubbed by each other in turn, and produce the never- failing change from generation to destruction.

6. Concretion of the Elements

BY MAKING USE OF THESE THE DEITY put together this world, sensible to touch through the particles of Earth, and to sight through those of Fire, which two are the extremes. Through the particles of Air and Water he had conjoined the world by the strongest chain, namely proportion, which restrains not only itself but all its subjects. Now if the conjoined object is a plane surface one middle term is sufficient, but if a solid there will be need of two mean terms. With two middle terms, therefore, he combined two extremes, so that as Fire is to Air, Air is to Water; and as Air is to Water, so is Water to Earth; and by alternation, as Fire is to Water, Air might be to Earth. Now since all are equal in power, their ratios are in a state of equilibrium. This world then is one, through the bond of the deity, made according to proportion.

Now each of these substances possesses many forms. Fire has those of flame, burning and luminousness, through the inequality of the triangles in each of them. In the same manner Air is partly clear and dry, and partly turbid and foggy. Water can be partly flowing and partly congealed, as it is in snow, frost, hail or ice. That which is moist is in one respect flowing as honey and oil, but in another is compact as pitch and wax. Concerning solids some are fusible, as gold, silver, copper, tin, lead and copper, and some brittle as sulphur, asphalt, nitre, salt, alum, and similar materials.

7. Composition of the Soul

AFTER PUTTING TOGETHER THE WORLD, the deity planned the creation of mortal beings so that, himself being perfect, he might perfectly complete the world. Therefore he mixed up the soul of man out of the same proportions and powers, and after taking the particles and distributing them, he delivered them over to Nature, whose office is to effect change. She then took up the task of working out mortal and ephemeral living beings, whose souls were drawn in from different sources, some from the moon, others from the sun, and others from various planets, [from] that cycle within the Difference, with the exception of one single power which was derived from Sameness, which she mixed up in the rational portion of the soul, as the image of wisdom in those of a happy fate.

Now in the soul of man one portion is rational and intellectual, and another irrational and unintellectual. Of the logical part the best portion is derived from Sameness, while the worst comes from Difference, and each is situated around the head so that the other parts of the soul and body may minister to it, as the supreme part of the whole body. Of the irrational portion, that which represents passion hovers around the heart, while desire inhabits the liver. The principle of the body and root of the marrow is the brain, wherein resides the ruling power; and from this, like an effusion, through the back-bone flows what is left over from the brain, from which are separated the particles of semen and seed. The marrow's surrounding defenses are the bones, of which the flesh is the covering and concealment. To the nerves he united joints by tendons, suitable for their movement. Of the internal organs, some exist for the sake of nourishment and others for safety. Of exterior motions, some are conveyed to the interior intelligent places of perception while others, not falling under the power of apprehension, are unperceived, either because the affected bodies are too earth-like or because the movements are too feeble. The painful movements tend to arouse nature, while the pleasurable lull nature into remaining within herself.

8. Sensations

AMONG THE SENSES, the deity has in us lit sight to view the objects in the heavens and for the reception of knowledge, while to make us capable to receive speech and melody, he has implanted hearing in us, of which he who is deprived thereof from birth will become dumb, nor be able to utter any speech, and that is why this sense is said to be related closest to speech.

Many affections of the body that have a name are so called with reference to touch, and others from relation to their seat. Touch judges the properties connected with life such as warmth, coldness, dryness, moisture, smoothness, roughness, and of things that are yielding, opposing, hard or soft. Touch also judges heaviness or lightness. Reason defines these affections as being centripetal and centrifugal, which men mean to express when they say below and middle. For the center of a sphere is below, and that part lying above it and stretching to the circumference is called upwardness.

Now what is warm appears to consist of fine particles, causing bodies to separate, while coldness consists of the grossness of the particles, causing a tendency to condense.

The circumstances connected with the sense of taste are similar to those of touch. For substances grow either astringent or smooth through contraction and dilation, also by entering the pores and assuming shapes. For those that cause the tongue to melt away, or that scrape it, appear to be rough. Those that act moderately in scraping appear brackish. Those that inflame or separate the skin are acrid; while their opposites, the smooth and sweet, are reduced to a juicy state.

The kinds of odors have not been defined because they percolate through narrow pores that are too stiff to be closed or separated. Things seem to be sweet smelling or bad smelling from the putrefaction or concoction of the earth and of similar substances.

A vocal sound is a percussion in the air, arriving at the soul through the ears. The passages of the ears reach to the liver, and among them is pneuma, by the movement of which hearing exists. Now of the voice and hearing, that portion which is quick is acute, while that which is slow is grave, the medium being the most harmonious. What is much and diffused is great; what is little and compressed is small; what is arranged to musical proportions is in tune; while that which is unarranged and unproportionate is out of tune and not properly adjusted.

The fourth kind of things relating to the senses is the most uniform and various, and they are called objects of sight, in which are all kinds of colors and an infinity of colored substances. The principle ones are four: white. black, brilliant [blue] and red, out of a mixture of which all other colors are prepared. What is white dilates the eye, and what is black causes it to contract, just as warmth expands, and cold contracts, and what is rough contracts the tasting, and what is sharp dilates it.

9. Respiration

IT IS NATURAL for the covering of animals that live in the air to be nourished and kept together by the food being distributed in the veins through the whole mass, in the manner of a stream, conveyed as it were by channels, and moistened by the breath, which diffuses it, and carries it to the extremities. Respiration is produced because there is no vacuum in nature: the air, as it flows in, is inhaled in place of that which is exhaled through unseen pores, such as those through which perspiration drops appear on the skin, but a portion is excreted by the natural warmth of the body. It then becomes necessary for an equivalent portion to be reintroduced to avoid a vacuum.

Now in lifeless substances, according to the analogy of respiration, the same organization occurs. The cupping-glass and amber, for instance, bear resemblance to respiration. [2]

Now the breath flows through the body to an orifice outwards and is, in turn, introduced through respiration by the mouth and nostrils, and again after the manner of the flow of the Euripus, is in turn carried to the body which is extended according to the expiration. Also the cupping-glass, when the air within is expelled by fire, attracts moisture to itself; and amber, when the air is separated from it, attracts a nearby substance.

Now all nourishment comes as from a root from the heart, and from the stomach as a fountain, and is conveyed to the body to which, if it be moistened by more than what flows out, there is said to be an increase, but if more flows out it is known as a decay. The point of perfection is the boundary between these two, and is considered to exist in an equality of the efflux and influx; but when the joints of the system are broken, should there no longer exist any passage for the breath, or the nourishment should not be distributed, then the animal dies.

10. Disorders

THERE ARE MANY THINGS HURTFUL TO LIFE which are the causes of death. One kind is disease. Its beginning is disharmony of the functions when the simple powers such as heat, cold, moisture or dryness are excessive or deficient. Then come changes and alterations in the blood from corruption and the deterioration of the flesh, when changes in the blood or flesh take place according to the changes of what is acid, brackish, or pungent in the blood. Hence arise the production of bile and of phlegm, of diseased juices, and of rotten liquids. Their effects are weak indeed, unless deeply seated, but difficult to cure when their commencement is generated from the bones, and acute if located in the marrow. The last disorders are those of the breath, bile and phlegm, when they increase and flow into places inappropriate for them. For by taking the place of the better, and driving away what is congenial, they fix themselves there, injuring bodies, and resolving these into themselves.

These then are the sufferings of the body, and hence arise many diseases of the soul, some from one faculty, some from another. Of the perceptive soul the disease is a difficulty of perception; of the recollecting, a forgetfulness; of the appetitive part, a deficiency of desire and eagerness; of the affective, a violent suffering and excited madness; of the rational, an indisposition to learn and think.

But of wickedness the beginnings are pleasures and pains; desires and fears, inflamed by the body, mingled with the mind, and are called by different names. For there loves and desires let loose uncontrolled passions, heavy resentments, and appetites of various kinds, and immoderate pleasures. Plainly, to be unreasonably disposed towards the affections is the limit of virtue, and to be under their rule is that of vice; for to abound in them, or to be superior to them, places us in a good or bad position. Against such impulses the temperaments of our bodies are greatly able to cooperate whether quick or hot, or various, by leading us to melancholy or violent lewdness; and certain parts, when affected by a flux, produce itchings and forms of body more similar to a state of inflammation than one of health, through which a sinking of the spirits and a forgetfulness, delirium and a state of fear result.

11. Discipline

IMPORTANT, TOO, ARE THE HABITS in which persons are trained, in the city or at home, and their daily food, by luxury enervating the soul, or fortifying it for strength. For living out of doors, and simple fare, and gymnastic exercises, and the morals of companions, produce the greatest effect in the way of vice and virtue. These causes are derived from our parents and the elements, rather than ourselves, provided that on our part there be no remissness by keeping aloof from acts of duty. The animal cannot be in good condition unless the body possesses the better properties under its control, namely health and correct perception, and strength and beauty. Now the principles of beauty are a symmetry as regards its parts, and as regards the soul. For nature has arranged the body, like an instrument, to be subservient to and in harmony with the subjects of life. The soul must likewise be brought into harmony with its analogous good qualities, namely in the case of temperance, as the body is in the case of health; and in that of prudence, as in the case of correct perception; and in that of fortitude, as in the case of vigor and strength; and in that of justice, as in the case of beauty.

Nature, of course, furnishes their beginnings, but their continuation and maturation result from carefulness: those relating to the body from the gymnastic and medical arts, those to the soul through instruction and philosophy. For these are the powers that nourish and give a tone to the body and soul by means of labor and gymnastic exercise and pureness of diet; some through drug medication applied to the body, and others through discipline applied to the soul by means of punishments and reproaches, for by the encouragement they give strength and excite to an onward movement and exhort to beneficial deeds. The art of the gymnasium trainer and its nearest approximation, that of the medical man, do on application to the body reduce their powers to the utmost symmetry, purifying the blood and equalizing the breath so that, if there were any diseased virulence there, the powers of blood and breath may be made vigorous; but music and its leader philosophy which the Gods ordained as regulators for the soul, accustom, persuade, and partly compel the irrational to obey reason, and induce the spirited and appetitive parts of the soul to become one mild and the other quiet, so as not to be moved without reason, nor to be unmoved when the mind incites either to desire or enjoy something; for this is the definition of temperance, namely, docility and firmness. Intelligence and philosophy the highest in honor, after cleansing the soul from false opinions, have introduced knowledge, recalling the mind from excessive ignorance and setting it free for the contemplation of divine things, in which to occupy oneself with self-sufficiency, as regards the affairs of a man, and with an abundance, for the commensurate period of life, is a happy state.

12. Human Destiny

NOW HE WHOM THE deity has happened to assign somewhat of a good fate is, through opinion, led to the happiest life. But if he be morose and indocile, let the punishment that comes from law and reason follow him, bringing with it the fears ever on the increase, both those that originate in heaven or Hades, how that punishments inexorable are below laid up for the unhappy, as well as those ancient Homeric threats of retaliation for the wickedness of those defiled by crime. For as we sometimes restore bodies to health by means of diseased substances, if they will not yield to the more healthy, so if the soul will not be led by true reasoning, we restrain it by false. These are unusual since, by a change, we say that the souls of cowards enter into the bodies of women who are inclined to insulting conduct; and the souls of the blood-stained take on the bodies of wild beasts; the lascivious enter into the bodies of sows and boars; the light-minded and frivolous take on shapes of aeronautic birds; and those who neither learn nor think of anything enter into the bodies of idle fish.

On all these matters, however, there has at a second period been delivered a judgment by Nemesis, of Fate, together with the avenging deities that preside over murderers, and those under the earth in Hades, and the inspectors of human affairs to whom God, the leader of all, has entrusted the administration of the world, which is filled with Gods and men, and the rest of the living beings who have been fashioned according to the best model of an unbegotten, eternal and mentally-perceived form.

_______________

Notes:

1. Plato identified the power of Sameness with the regular movement of the stars and the celestial sphere. The power of Difference is identified with the movements of the planets which, over the course of the year, wander in the opposite direction of the fixed stars.

2. As Thomas H. Tobin observes in his translation of this text, the cupping-glass was a device used in medicine. When heated the air inside was expelled and the device was placed on a wound to remove poison. Rubbing amber with a cloth was likewise thought to expell the air in it, causing the amber to attract small object to it as the vacuum was filled. The actual cause is static electricity.

[admin]

SELECT PASSAGES FROM THE CHURCH FATHERS

IT WAS NOT UNCOMMON for the early church fathers to refer to the philosophers of Greece, even though for many the contact with their thought was only through outlines and handbooks rather than through primary sources.

For the most part the church fathers had positive things to say about the Pythagoreans. Clement of Alexandria, who had more than a passing acquaintance with Platonic thought, shows an interest in Pythagorean mathematics, and even applied the numbers of the harmonic ratio to the interpretation of scripture. Moreover, Justin Martyr sought entrance to a Pythagorean school but was rejected on account of inadequate mathematical knowledge; he later became a Platonist and then a Christian. The fragments which follow, which were not included in Guthrie's original edition of this book, are representative of what the church fathers had to say.

NOTICES FROM THE CHURCH FATHERS

HERACLEIDES AND THE PYTHAGOREANS think: that each of the stars is a world, including an Earth, and an atmosphere and an ether in the infinite space. These doctrines are introduced in the Orphic Hymns, for they make each star a world. -- Ps.-Plutarch, On the Opinions of the Philosophers. Quoted by Eusebius, Preparation for the Gospel, 839b.

Then, in regular succession from another starting-point, Pythagoras the Samian, son of Mnesarchus, calls numbers, with their proportions and harmonies, and the elements composed of both, the first principles; and he includes also Unity and the Indefinite Dyad. - Justin Martyr, Exhortatory Address to the Greeks, IV.

For the Pythagorean Theano writes, "Life would indeed be a feast to the wicked, who, having done evil, then die; were not the soul immortal, death would be a godsend." -- Clement of Alexandria, Stromateis IV, 7.

Did not Theano the Pythagorean make such progress in philosophy, that to him who looked intently at her, and said, "Your arm is beautiful," she answered "Yes, but it is not public." Characterized by the same propriety there is also reported the following reply. When asked when a woman after being with her husband may attend a religious festival, said, "From her own husband at once, from a stranger never." -- Clement of Alexandria, Stromateis, IV, 19.

Pythagoras thus defined the being of God, "as a soul passing to and fro, and diffused through all parts of the universe, and through all nature, from which all living creatures which are produced derive their life." -- Lactantius, The Divine Institutes, I, 5.

And Pythagoras, son of Mnesarchus, who expounded the doctrines of his own philosophy mystically by means of symbols, as those who have written his life show, himself seems to have entertained thoughts about the unity of God not unworthy of his foreign residence in Egypt. For when he says that Unity is the first principle of all things, and that it is the cause of all good, he teaches by an allegory that God is one, and alone. And that this is so, is evident from his saying that unity and one differ widely from one another. For he says that unity belongs to the class of things perceived by the mind, but that one belongs to numbers. And if you desire to see a clearer proof of the opinion of Pythagoras concerning one God, hear his own opinion, for he spoke as follows: "God is one; and he himself does not, as some suppose, exist outside the world, but in it, he being wholly present in the entire circle, and beholding all generations, being the regulating ingredient of all the ages, and the administrator of his own powers and works, the first principle of all things, the light of heaven, and father of all, the intelligence and animating soul of the universe, the movement of all orbits." Thus, then, Pythagoras. -- Justin Martyr, Exhortatory Address to the Greeks, XIX.

[admin]

PASSAGES REFERRING TO THE PYTHAGOREANS

FROM PLATO AND ARISTOTLE

TRANSLATED BY ARTHUR FAIRBANKS

PASSAGES IN PLATO REFERRING TO THE PYTHAGOREANS

Phaedo 62 B. The saying uttered in secret rites, to the effect that we men are in a sort of prison, and that one ought not to release himself from it nor yet to run away, seems to me something great and not easy to see through; but this at least I think is well said, that it is the Gods who care for us, and we men are one of the possessions of the Gods.

Crarylus 400 B. For some say that it [the body] is the tomb of the soul -- I think it was the followers of Orpheus in particular who introduced this word -- which has the soul enclosed like a prison in order that it may be kept safe.

Gorgias 493 A. I once heard one of the wise men say that now we are dead and the body (soma) is our tomb (sema), and that the part of the soul where desires are, it so happens, is open to persuasion, and moves upward or downward. And, indeed, a clever man -- perhaps some inhabitant in Sicily or Italy -- speaking allegorically, and taking the word from 'credible' (pithanos) and 'persuadable' (pistikos), called this a jar (pithos); and he called those without intelligence uninitiated, and that part of the soul of uninitiated persons where the desires are, he called its intemperatness, and said it was not water-tight, as a jar might be pierced with holes -- using the simile because of insatiate desires.

Gorgias 507 E. And the wise men say that one community embraces heaven and earth and Gods and men and friendship and order and temperance and righteousness, and for that reason they call this whole a kosmos, my friend, for it is not without order nor yet is there excess. It seems to me that you do not pay attention to these things, though you are wise in regard to them. But it has escaped your notice that geometrical equality prevails widely among both Gods and men.

PASSAGES IN ARISTOTLE REFERRING TO THE PYTHAGOREANS

Physics. iii. 4; 203 a I. Some, like the Pythagoreans and Plato, have made the Unlimited a first principle existing by itself, not connected with anything else, but being the infinite itself in its essence. Only the Pythagoreans found it among all things perceived by sense (for they say that number is not an abstraction), and they held that what is outside the heavens is Unlimited.

iii. 4; 203 a 10. The Pythagoreans identify the Unlimited with the even, For this, they say, is cut off and shut in by the odd, and provides things with an element of infinity. An indication of this is what happens with numbers. If gnomons are place round the one, and without the one, in the one construction the figure that results is always the same [square], in the other it is always different [oblong].

iii. 4; 204 a 33. [The Pythagoreans] both hold that the infinite is substance, and divide it into parts.

iv. 6; 213 b 22. And the Pythagoreans say that there is a void, and that it enters into the heaven itself from the infinite air, as though it [the heaven] were breathing; and this void defines the natures of things, inasmuch as it is a certain separation and definition of things that lie together; and this is true first in the case of numbers, for the void defines the nature of these.

On the Heavens. i. 1; 268 a 10. For as the Pythagoreans say, the All and all things are defined by threes; for end and middle and beginning constitute the number of the All, and also the number of the Triad.

ii. 2; 284 b 6. And since there are some who say that there is a right and a left of the heavens, as, for instance, those that are called Pythagoreans (for such is their doctrine), we must investigate whether it is as they say.

ii. 2; 285 a 10. Wherefore one of the Pythagoreans might be surprised in that they say that there are only these two first principles, the right and the left, and they pass over four of them as not having the least validity; for there is no less difference up and down, and front and back than there is right and left in all creatures.

ii. 2; 285 b 23. And some are dwelling in the upper hemisphere and to the right, while we dwell below and to the left, which is the opposite to what the Pythagoreans say; for they put us above and to the right, while the others are below and at the left.

ii. 9; 290 b 15. Some think it necessary that noise should arise when so great bodies are in motion, since sound does arise from bodies among us which are not so large and do not move so swiftly; and from the sun and moon and from the stars in so great number, and of so great size, moving so swiftly, there must necessarily arise a sound inconceivably great. Assuming these things and that the swiftness has the principle of harmony by reason of the intervals, they say that the sound of the stars moving on in a circle becomes musical. And since it seems unreasonable that we also do not hear this sound, they say that the reason for this is that the sound exists in the very nature of things, so as not to be distinguishable from the opposite silence; for the distinction of sound and silence lies in their contrast with each other, so that as blacksmiths think there is no difference between them because they are accustomed to the sound, so the same thing happens to men.

ii. 9; 291 a 7. What occasions the difficulty and makes the Pythagoreans say that there is a harmony of the bodies as they move, is a proof. For whatever things move themselves make a sound and noise; but whatever things are fastened in what moves or exist in it as the parts in a ship, cannot make a noise, nor yet does the ship if it moves in a river.

ii. 12; 293 a 19. They say that the whole heaven is limited, the opposite to what those of Italy, called the Pythagoreans, say; for these say that fire is at the center and that the earth is one of the stars, and that moving in a circle about the center it produces night and day. And they assume yet another earth opposite this which they call the counter-earth (antichthon), not seeking reasons and causes for phenomena, but stretching phenomena to meet certain assumptions and opinions of theirs and attempting to arrange them in a system....And what is more, the Pythagoreans say that the most authoritative part of the All stands guard, because it is specially fitting that it should, and this part is the center; and this place that the fire occupies, they call the Guardpost of Zeus, as it is called simply the center, that is, the center of space and the center of matter and of nature.

Metaphysics. i. 5; 985 b 23-986 b 8. With these before them [Anaxagoras, Empedocles, Atomists] those called Pythagoreans, applying themselves to the sciences, first developed them; and being brought up in them they thought that the first principles of these [i.e., numbers] were the first principles of all things. And since of these [sciences] numbers are by nature the first, in numbers rather than in fire and earth and water they thought they saw many likenesses to things that are and that are coming to be, as, for instance, justice is such a property of numbers, and soul and mind are such a property, and another is opportunity, and of other things one may say the same of each one.

And further, discerning in numbers the conditions and reasons of harmonies also -- since, moreover, other things seemed to be like numbers in their entire nature, and numbers were the first of every nature -- they assumed that the elements of numbers were the elements of all things, and that the whole heavens were harmony and number. And whatever characteristics in numbers and harmonies they could show were in agreement with the properties of the heavens and its parts and with its whole arrangement, these they collected and adapted; and if there chanced to be any gap anywhere, they eagerly sought that the whole system might be connected with these [stray phenomena]. To give an example of my meaning: inasmuch as ten seemed to be the perfect number and to embrace the whole nature of numbers, they asserted that the number of bodies moving through the heavens were ten, and when only nine were visible, for the reason just stated they postulated the counter-earth as the tenth. We have given a more definite account of these thinkers in other parts of our writings. But we have referred to them here with this purpose in view, that we might ascertain from them what they asserted as the first principles and in what manner they came upon the causes that have been enumerated. They certainly seem to consider Number as the first principle and, as it were, the matter in things and in their conditions and states; and the odd and the even are elements of number, and of these the one is Limited and the other Unlimited, and unity is the product of both of them, for it is both odd and even, and Number arises from the one, and the whole heaven, as has been said, is numbers.

A different party in this same school says that the first principles are ten, named according to the following table: Limited and Unlimited, Odd and Even, One and Many, Right and Left, Male and Female, Rest and Motion, Straight and Crooked, Light and Darkness, Good and Bad, Square and Oblong. After this manner Alcmaeon of Croton seems to have conceived them, and either he received this doctrine from them or they from him, for Alcmaeon arrived at maturity when Pythagoras was an old man, and his teachings resembled theirs. For he says that most human affairs are twofold, not meaning opposites reached by definition, as did the former party, but opposites by chance -- as, for example, white-black, sweet-bitter, good-bad, small-great. This philosopher let fall his indefinite opinions about the other contraries, but the Pythagoreans declared the number of the opposites and what they were. From both schools one may learn this much: that opposites are the first principles of things -- but from the latter he may learn the number of these, and what they are. Yet how it is possible to bring them into relation with the causes of which we have spoken they have not clearly worked out. They seem to range their elements under the category of matter, for they say that substance is compounded and formed from them, and that they inhere in it.

987 a 9-27. Down to the Italian philosophers, and with their exception, the rest have spoken more reasonably about these principles, except that, as we said, they do indeed use two principles, and the one of these, whence is motion, some regard as one and others as twofold. The Pythagoreans, however, while they in similar manner assume two first principles, add this which is peculiar to themselves: that they do not think that the Limited and the Unlimited and the One are certain other things by nature, such as fire or earth or any other such thing, but the Unlimited itself and Unity itself are the essence of things of which they are predicated, and so they make Number the essence of all things. So they taught after this manner about them, and began to discourse and to define what essence is, but they made it altogether too simple a matter. For they made their definition superficially, and to whatever first the definition might apply, this they thought to be the essence of the matter, as if one should say that twofold and two were the same, because the twofold subsists in the two. But undoubtedly the two and the twofold are not the same, otherwise one thing will be many -- a consequence which they actually drew. So much then may be learned from the earlier philosophers and from their successors.

i. 6; 987 b 10. And Plato only changed the name, for the Pythagoreans say that things exist by 'imitation' of numbers, but Plato, by 'participation.'

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

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

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

iii. 4; 1001 a 9. Plato and the Pythagoreans assert that neither being Iior unity is something different from things, but that it is the very nature of them, as though essence itself consisted in unity and existence.

vii. 10; 1036 b 17. So it turns out that many things of which the forms appear different have one Form, as the Pythagoreans discovered; and one can say that there is one Form for everything, and hold that others are not forms, and thus all things will be one.

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

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

xiii. 4; 1078 b 21. The Pythagoreans [before Democritus] only defined a few things, the concepts of which they reduced to numbers, as for instance opportunity or justice or marriage...

xii. 6; 1080 b 16. The Pythagoreans say that there is but one number, the mathematical, but that things of sense are not separated from this, for they are composed of it; indeed, they construct the whole heaven out of numbers, but not out of abstract numbers, for they assume that the units have magnitude; but how the first unit was so constituted to have magnitude they seem at a loss to say.

xiii. 6; 1080 b 31. All, as many as regard the one as the element and first principle of things, except the Pythagoreans, assert that numbers are based on the unit; but the Pythagoreans assert, as has been remarked, that numbers have magnitude.

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

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

Nicomachean Ethics. i. 6; 1096 b 5. And the Pythagoreans seem to speak more persuasively about it, putting unity in the column of good things.

ii. 6; 1106 b 29. Evil partakes of the nature of the Unlimited, Good of the Limited, as the Pythagoreans conjectured.

v. 5; 1132 b 21. Reciprocity seems to some to be absolutely just, as the Pythagoreans say; for these defined the just as that which is reciprocal to another.

Moralia. i. 1; 1182 a 11. First Pythagoras attempted to speak concerning virtue, but he did not speak correctly; for bringing virtues into correspondence with numbers, he did not make any distinct.

[admin]

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



_______________

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.

[admin]

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.

_______________

Notes:

1. Plato, Timaeus 36BC.

[admin]

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

[admin]

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.

[admin]

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.

_______________

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.

[admin]

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.

[admin]

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.