(129) And of the Solar Circles, and of the Lunar, clashings, and of the Aërial Recesses; the Melody of Ether, and of the Sun, and of the phases of the Moon, and of the Air.

(1) Since, however, we are narrating the wisdom employed by Pythagoras in instructing his disciples, it will not be unappropriate to relate that which is proximate in a following order to this, viz. how he invented the harmonic science, and harmonic ratios. But for this purpose we must begin a little higher. Intently considering once, and reasoning with himself, whether it would be possible to devise a certain instrumental assistance to the hearing, which should be firm and unerring, such as the sight obtains through the compass and the rule, or, by Jupiter, through a dioptric instrument; or such as the touch obtains through the balance, or the contrivance of measures;—thus considering, as he was walking near a brazier’s shop, he heard from a certain divine casualty the hammers beating out a piece of iron on an anvil, and producing sounds that accorded with each other, one combination only excepted.

But he recognized in those sounds, the diapason, the diapente, and the diatessaron, harmony. He saw, however, that the sound which was between the diatessaron and the diapente was itself by itself dissonant, yet, nevertheless, gave completion to that which was the greater sound among them. Being delighted, therefore, to find that the thing which he was anxious to discover had succeeded to his wishes by divine assistance, he went into the brazier’s shop, and found by various experiments, that the difference of sound arose from the magnitude of the hammers, but not from the force of the strokes, nor from the figure of the hammers, nor from the transposition of the iron which was beaten. When, therefore, he had accurately examined the weights and the equal counterpoise of the hammers, he returned home, and fixed one stake diagonally to the walls, lest if there were many, a certain difference should arise from this circumstance, or in short, lest the peculiar nature of each of the stakes should cause a suspicion of mutation.

Afterwards, from this stake he suspended four chords consisting of the same materials, and of the same magnitude and thickness, and likewise equally twisted. To the extremity of each chord also he tied a weight. And when he had so contrived, that the chords were perfectly equal to each other in length, he afterwards alternately struck two chords at once, and found the before-mentioned symphonies, viz. a different symphony in a different combination. For he discovered that the chord which was stretched by the greatest weight, produced, when compared with that which was stretched by the smallest, the symphony diapason. But the former of these weights was twelve pounds, and the latter six. And, therefore, being in a duple ratio, it exhibited the consonance diapason; which the weights themselves rendered apparent.

But again, he found that the chord from which the greatest weight was suspended compared with that from which the weight next to the smallest depended, and which weight was eight pounds, produced the symphony diapente. Hence he discovered that this symphony is in a sesquialter ratio, in which ratio also the weights were to each other. And he found that the chord which was stretched by the greatest weight, produced, when compared with that which was next to it in weight, and was nine pounds, the symphony diatessaron, analogously to the weights. This ratio, therefore, he discovered to be sesquitertian; but that of the chord from which a weight of nine pounds was suspended, to the chord which had the smallest weight [or six pounds,] to be sesquialter.

For 9 is to 6 in a sesquialter ratio. In like manner, the chord next to that from which the smallest weight depended, was to that which had the smallest weight, in a sesquitertian ratio, [for it was the ratio of 8 to 6,] but to the chord which had the greatest weight, in a sesquialter ratio [for such is the ratio of 12 to 8.] Hence, that which is between the diapente and the diatessaron, and by which the diapente exceeds the diatessaron, is proved to be in an epogdoan ratio, or that of 9 to 8. But either way it may be proved that the diapason is a system consisting of the diapente in conjunction with the diatessaron, just as the duple ratio consists of the sesquialter and sesquitertian, as for instance, 12, 8, and 6; or conversely, of the diatessaron and the diapente, as in the duple ratio of the sesquitertian and sesquialter ratios, as for instance 12, 9, and 6.

After this manner, therefore, and in this order, having conformed both his hand and his hearing to the suspended weights, and having established according to them the ratio of the habitudes, he transferred by an easy artifice the common suspension of the chords from the diagonal stake to the limen of the instrument, which he called chordotonon . But he produced by the aid of pegs a tension of the chords analogous to that effected by the weights.

(617) coloured by the reflected light of the seventh; the second and fifth [Saturn and Mercury] are in colour like one another, and yellower than the preceding; the third [Venus] has the whitest light; the fourth [Mars] is reddish; the sixth [Jupiter] is in whiteness second. Now the whole spindle has the same motion; but, as the whole revolves in one direction, the seven inner circles move slowly in the other, and of these the swiftest is the eighth; next in swiftness are the seventh, sixth, and fifth, which move together; third in swiftness appeared to move according to the law of this reversed motion the fourth; the third appeared fourth and the second fifth. The spindle turns on the knees of Necessity; and on the upper surface of each circle is a siren, who goes round with them, hymning a single tone or note. The eight together form one harmony; and round about, at equal intervals, there is another band, three in number, each sitting upon her throne: these are the Fates, daughters of Necessity, who are clothed in white robes and have chaplets upon their heads, Lachesis and Clotho and Atropos, who accompany with their voices the harmony of the sirens—Lachesis singing of the past, Clotho of the present, Atropos of the future; Clotho from time to time assisting with a touch of her right hand the revolution of the outer circle of the whorl or spindle, and Atropos with her left hand touching and guiding the inner ones, and Lachesis laying

(530) them are obvious enough even to wits no better than ours; and there are others, as I imagine, which may be left to wiser persons. But where are the two? There is a second, I said, which is the counterpart of the one already named. And what may that be? The second, I said, would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions; and these are sister sciences—as the Pythagoreans say, and we, Glaucon, agree with them? Yes, he replied. But this, I said, is a laborious study, and therefore we had better go and learn of them; and they will tell us whether there are any other applications of these sciences. At the same time, we must not lose sight of our own higher object. What is that? There is a perfection which all knowledge ought to reach, and which our pupils ought also to attain, and not to fall short of, as I was saying that they did in astronomy. For in the science of harmony, as you probably know, the same thing happens. The teachers of harmony compare the sounds and consonances which are heard only, and their labour, like that of the astronomers, is in vain. Yes, by heaven! he said; and ’tis as good as a play to hear them talking about their condensed notes, as they call them; they put their ears close alongside of the strings like persons catching a sound from their neighbour’s wall 5 —one set of them declaring that they distinguish an intermediate note and have found the least interval which should be the unit of measurement; the others insisting that the two sounds have passed into the same—either party setting

(2) Employing this method, therefore, as a basis, and as it were an infallible rule, he afterwards extended the experiment to various instruments; viz. to the pulsation of patellæ or pans, to pipes and reeds, to monochords, triangles, and the like. And in all these he found an immutable concord with the ratio of numbers. But he denominated the sound which participates of the number 6 hypate : that which participates of the number 8 and is sesquitertian, mese ; that which participates of the number 9, but is more acute by a tone than mese, he called paramese , and epogdous ; but that which participates of the dodecad, nete . Having also filled up the middle spaces with analogous sounds according to the diatonic genus, he formed an octochord from symphonious numbers, viz. from the double, the sesquialter, the sesquitertian, and from the difference of these, the epogdous.

And thus he discovered the [harmonic] progression, which tends by a certain physical necessity from the most grave [i. e. flat] to the most acute sound, according to this diatonic genus. For from the diatonic, he rendered the chromatic and enharmonic genus perspicuous, as we shall some time or other show when we treat of music. This diatonic genus, however, appears to have such physical gradations and progressions as the following; viz. a semitone, a tone, and then a tone; and this is the diatessaron, being a system consisting of two tones, and of what is called a semitone. Afterwards, another tone being assumed, viz. the one which is intermediate, the diapente is produced, which is a system consisting of three tones and a semitone.

In the next place to this is the system of a semitone, a tone, and a tone, forming another diatessaron, i. e. another sesquitertian ratio. So that in the more ancient heptachord indeed, all the sounds, from the most grave, which are with respect to each other fourths, produce every where with each other the symphony diatessaron; the semitone receiving by transition, the first, middle, and third place, according to the tetrachord. In the Pythagoric octachord, however, which by conjunction is a system of the tetrachord and pentachord, but if disjoined is a system of two tetrachords separated from each other, the progression is from the most grave sound. Hence all the sounds that are by their distance from each other fifths, produce with each other the symphony diapente; the semitone successively proceeding into four places, viz. the first, second, third, and fourth. After this manner, therefore, it is said that music was discovered by Pythagoras. And having reduced it to a system, he delivered it to his disciples as subservient to every thing that is most beautiful.

(66) This opinion or supposition is not right, but the earth rolleth itself about; and runneth with the other planets, as in a wheel, round about the sun. The earth does not remain staying in one place, but in a year runneth round once about the sun, as the other planets next the sun, but Saturn and Jupiter, as also Mars, by reason of their great orb, circumference, and great height, cannot do it [in a year], because they stand so high above, and far distant from the SUN. Now it may be asked, What is the SUN, and what are the other PLANETS? Or how are they come to be?

(530) other proportion. No, he replied, such an idea would be ridiculous. And will not a true astronomer have the same feeling when he looks at the movements of the stars? Will he not think that heaven and the things in heaven are framed by the Creator of them in the most perfect manner? But he will never imagine that the proportions of night and day, or of both to the month, or of the month to the year, or of the stars to these and to one another, and any other things that are material and visible can also be eternal and subject to no deviation—that would be absurd; and it is equally absurd to take so much pains in investigating their exact truth. I quite agree, though I never thought of this before. Then, I said, in astronomy, as in geometry, we should employ problems, and let the heavens alone if we would approach the subject in the right way and so make the natural gift of reason to be of any real use. That, he said, is a work infinitely beyond our present astronomers. Yes, I said; and there are many other things which must also have a similar extension given to them, if our legislation is to be of any value. But can you tell me of any other suitable study? No, he said, not without thinking. Motion, I said, has many forms, and not one only; two of

(527) And suppose we make astronomy the third—what do you say? I am strongly inclined to it, he said; the observation of the seasons and of months and years is as essential to the general as it is to the farmer or sailor. I am amused, I said, at your fear of the world, which makes you guard against the appearance of insisting upon useless studies; and I quite admit the difficulty of believing that in every man there is an eye of the soul which, when by other pursuits lost and dimmed, is by these purified and re-illumined; and is more precious far than ten thousand bodily eyes, for by it alone is truth seen. Now there are two classes of persons: one class of those who will agree with you and will take your words as a revelation; another class to whom they will be utterly unmeaning, and who will naturally deem them to be idle tales, for they see no sort of profit which is to be obtained from them. And therefore you had better decide at once with which of the two you are proposing to argue. You will very likely say with neither, and that your chief aim in carrying on the argument is your own improvement; at the same time you do not grudge to others any benefit which they may receive. I think that I should prefer to carry on the argument mainly on my own behalf. Then take a step backward, for we have gone wrong in the order of the sciences. What was the mistake? he said. After plane geometry, I said, we proceeded at once to

(9) Now the Lord, having come alone into the intellectual world, enters by His sufferings, introduced into the knowledge of the Ineffable, ascending above every name which is known by sound. The lamp, too, was placed to the south of the altar of incense; and by it were shown the motions of the seven planets, that perform their revolutions towards the south. For three branches rose on either side of the tamp, and lights on them; since also the sun, like the lamp, set in the midst of all the planets, dispenses with a kind of divine music the light to those above and to those below.

(15) The same holds also of astronomy. For treating of the description of the celestial objects, about the form of the universe, and the revolution of the heavens, and the motion of the stars, leading the soul nearer to the creative power, it teaches to quickness in perceiving the seasons of the year, the changes of the air, and the appearance of the stars; since also navigation and husbandry derive from this much benefit, as architecture and building from geometry. This branch of learning, too, makes the soul in the highest degree observant, capable of perceiving the true and detecting the false, of discovering correspondences and proportions, so as to hunt out for similarity in things dissimilar; and conducts us to the discovery of length without breadth, and superficial extent without thickness, and an indivisible point, and transports to intellectual objects from those of sense.

(1) It is also said, that Pythagoras was the first who called himself a philosopher; this not being a new name, but previously instructing us in a useful manner in a thing appropriate to the name. For he said that the entrance of men into the present life, resembled the progression of a crowd to some public spectacle. For there men of every description assemble with different views; one hastening to sell his wares for the sake of money and gain; but another that he may acquire renown by exhibiting the strength of his body; and there is also a third class of men, and those the most liberal, who assemble for the sake of surveying the places, the beautiful works of art, the specimens of valor, and the literary productions which are usually exhibited on such occasions.

Thus also in the present life, men of all-various pursuits are collected together in one and the same place. For some are influenced by the desire of riches and luxury; others by the love of power and dominion; and others are possessed with an insane ambition for glory. But the most pure and unadulterated character, is that of the man who gives himself to the contemplation of the most beautiful things, and whom it is proper to call a philosopher. He adds, that the survey of all heaven, and of the stars that revolve in it, is indeed beautiful, when the order of them is considered. For they derive this beauty and order by the participation of the first and the intelligible essence.

But that first essence is the nature of number and reasons [i. e. productive principles,] which pervades through all things, and according to which all these [celestial bodies] are elegantly arranged, and fitly adorned. And wisdom indeed, truly so called, is a certain science which is conversant with the first beautiful objects, and these divine, undecaying, and possessing an invariable sameness of subsistence; by the participation of which other things also may be called beautiful. But philosophy is the appetition of a thing of this kind. The attention therefore to erudition is likewise beautiful, which Pythagoras extended, in order to effect the correction of mankind.

(31) The term "periodicity" so often employed in connection with the subject of Rhythm, means "state of occurring or recurring at fixed intervals of time." Every phenomenal thing manifests periodicity, by reason of the presence and activity of the Principle of Rhythm. Every phenomenal thing has its own rhythmic beat, or measure of periodicity. All scientific investigation tends to corroborate the ancient occult axiom: "Everything beats time." A leading scientist has said: "Rhythm is a necessary characteristic of all motion. Given the co-existence everywhere of antagonistic forces—a postulate which is necessitated by our experience—and Rhythm is a necessary corollary. All motion alternates—be it the motion of planets in their orbits, or ethereal corpuscles in their undulations—be it the cadence of speech, or the rise and fall of prices—it became manifest that this perpetual reversal of motion between limits is inevitable." The atoms in their vibrations manifest Rhythm. The swing of the planets and the whirling of the earth manifest Rhythm. The rise and fall of the tides manifest Rhythm. The swing of the pendulum is interrupted Rhythm. Completed Rhythm is represented only by a completed revolution or circular movement—uninterrupted Rhythm always manifests as a complete movement in an orbit. But inasmuch as the centre between the two extremes is, itself, moving in response to a higher order of Rhythm, we see at last that all completed Rhythm manifests as a spiral—a circular movement which at the same time is moving forward.

(529) nothing of that sort is matter of science; his soul is looking downwards, not upwards, whether his way to knowledge is by water or by land, whether he floats, or only lies on his back. I acknowledge, he said, the justice of your rebuke. Still, I should like to ascertain how astronomy can be learned in any manner more conducive to that knowledge of which we are speaking? I will tell you, I said: The starry heaven which we behold is wrought upon a visible ground, and therefore, although the fairest and most perfect of visible things, must necessarily be deemed inferior far to the true motions of absolute swiftness and absolute slowness, which are relative to each other, and carry with them that which is contained in them, in the true number and in every true figure. Now, these are to be apprehended by reason and intelligence, but not by sight. True, he replied. The spangled heavens should be used as a pattern and with a view to that higher knowledge; their beauty is like the beauty of figures or pictures excellently wrought by the hand of Daedalus, or some other great artist, which we may chance to behold; any geometrician who saw them would appreciate the exquisiteness of their workmanship, but he would never dream of thinking that in them he could find the true equal or the true double, or the truth of any

(84) Thus far the power of the life reached forth from the sun, and not higher; but the lustre or shining thereof, which has its power also, reacheth even to the stars, and through the whole body of this world. But thou must understand this exactly, from whence these two Planets are come to be.

(1) Let him imagine, who would well conceive What now I saw, and let him while I speak Retain the image as a steadfast rock, The fifteen stars, that in their divers regions The sky enliven with a light so great That it transcends all clusters of the air; Let him the Wain imagine unto which Our vault of heaven sufficeth night and day, So that in turning of its pole it fails not; Let him the mouth imagine of the horn That in the point beginneth of the axis Round about which the primal wheel revolves,— To have fashioned of themselves two signs in heaven, Like unto that which Minos' daughter made, The moment when she felt the frost of death; And one to have its rays within the other, And both to whirl themselves in such a manner That one should forward go, the other backward; And he will have some shadowing forth of that True constellation and the double dance That circled round the point at which I was; Because it is as much beyond our wont, As swifter than the motion of the Chiana Moveth the heaven that all the rest outspeeds.

(39) Timaeus: In this wise and for these reasons were generated Night and Day, which are the revolution of the one and most intelligent circuit; and Month, every time that the Moon having completed her own orbit overtakes the Sun; and Year, as often as the Sun has completed his own orbit. Of the other stars the revolutions have not been discovered by men (save for a few out of the many); wherefore they have no names for them, nor do they compute and compare their relative measurements, so that they are not aware, as a rule,

(38) Timaeus: The Moon He placed in the first circle around the Earth, the Sun in the second above the Earth; and the Morning Star and the Star called Sacred to Hermes He placed in those circles which move in an orbit equal to the Sun in velocity, but endowed with a power contrary thereto; whence it is that the Sun and the Star of Hermes and the Morning Star regularly overtake and are overtaken by one another. As to the rest of the stars, were one to describe in detail the positions in which He set them, and all the reasons therefor,

(3) The planets of this chain of worlds are not identical in composition and nature with the earth; on the contrary there is a wide difference between the several planets in this respect. The earth is not the highest in development in this chain, but on the contrary is far down on the scale, although there are others still lower. The progress of the souls around this chain of worlds, however, is not merely like a circle in which the soul travels from the lowest to the highest, but is rather according to the plan of the spiral, in which the journey always returns to the starting point, but on a higher plane of activity.

(46) The whole body to the neck signifieth and is the round circle or sphere of the stars, as also the deep within or between the stars, wherein the planets and elements reign.

(2) Moreover, the lations of the stars approximate to the eternal circulations of the heavens, not only locally, but also in powers, and the irradiations of light. But these are moved conformably to the mandates of the celestial Gods. For the most pure, agile, and supreme part of the air, is adapted to be enkindled [ i. e. is most inflammable], so that when the Gods assent, it is immediately set on fire. And if some one thinks that certain effluxions of the celestial bodies are imparted to the air, his opinion will not be discordant with what is frequently effected by the divine art. The union, also, and sympathy of the universe, and the simultaneous motion of the most remote parts, as if they were near, and belonged to one animal, cause these signs to be sent from the Gods to men in the most luminous manner, primarily, indeed, through the heavens, but afterwards through the air.