(5) One day while meditating upon the problem of harmony, Pythagoras chanced to pass a brazier's shop where workmen were pounding out a piece of metal upon an anvil. By noting the variances in pitch between the sounds made by large hammers and those made by smaller implements, and carefully estimating the harmonies and discords resulting from combinations of these sounds, he gained his first clue to the musical intervals of the diatonic scale. He entered the shop, and after carefully examining the tools and making mental note of their weights, returned to his own house and constructed an arm of wood so that it: extended out from the wall of his room. At regular intervals along this arm he attached four cords, all of like composition, size, and weight. To the first of these he attached a twelve-pound weight, to the second a nine-pound weight, to the third an eight-pound weight, and to the fourth a six-pound weight. These different weights corresponded to the sizes of the braziers' hammers.

(6) Pythagoras thereupon discovered that the first and fourth strings when sounded together produced the harmonic interval of the octave, for doubling the weight had the same effect as halving the string. The tension of the first string being twice that of the fourth string, their ratio was said to be 2:1, or duple. By similar experimentation he ascertained that the first and third string produced the harmony of the diapente, or the interval of the fifth. The tension of the first string being half again as much as that of the third string, their ratio was said to be 3:2, or sesquialter. Likewise the second and fourth strings, having the same ratio as the first and third strings, yielded a diapente harmony. Continuing his investigation, Pythagoras discovered that the first and second strings produced the harmony of the diatessaron, or the interval of the third; and the tension of the first string being a third greater than that of the second string, their ratio was said to be 4:3, or sesquitercian. The third and fourth strings, having the same ratio as the first and second strings, produced another harmony of the diatessaron. According to Iamblichus, the second and third strings had the ratio of 8:9, or epogdoan.

(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.

(5) Such, then, is the style of the example in arithmetic. And let the testimony of geometry be the tabernacle that was constructed, and the ark that was fashioned, - constructed in most regular proportions, and through divine ideas, by the gift of understanding, which leads us from things of sense to intellectual objects, or rather from these to holy things, and to the holy of holies. For the squares of wood indicate that the square form, producing fight angles, pervades all, and points out security. And the length of the structure was three hundred cubits, and the breadth fifty, and the height thirty; and above, the ark ends in a cubit, narrowing to a cubit from the broad base like a pyramid, the symbol of those who are purified and tested by fire. And this geometrical proportion has a place, for the transport of those holy abodes, whose differences are indicated by the differences of the numbers set down below.

(89) This sieve is a mathematical device originated by Eratosthenes about 230 B.C. far the purpose of segregating the composite and incomposite odd numbers. Its use is extremely simple after the theory has once been mastered. All the odd numbers are first arranged in their natural order as shown in the second panel from the bottom, designated Odd Numbers. It will then be seen that every third number (beginning with 3) is divisible by 3, every fifth number (beginning with 5;) is divisible by 5, every seventh number (beginning with 7) is divisible by 7, every ninth number (beginning with 9) is divisible by 9, every eleventh number (beginning with 11) is divisible by 11, and so on to infinity. This system finally sifts out what the Pythagoreans called the "incomposite" numbers, or those having no divisor other than themselves and unity. These will be found in the lowest panel, designated Primary and Incomposite Numbers. In his History of Mathematics, David Eugene Smith states that Eratosthenes was one of the greatest scholars of Alexandria and was called by his admirers "the second Plato." Eratosthenes was educated at Athens, and is renowned not only for his sieve but for having computed, by a very ingenious method, the circumference and diameter of the earth. His estimate of the earth's diameter was only 50 miles less than the polar diameter accepted by modern scientists. This and other mathematical achievements of Eratosthenes, are indisputable evidence that in the third century before Christ the Greeks not only knew the earth to be spherical in farm but could also approximate, with amazing accuracy, its actual size and distance from both the sun and the moon. Aristarchus of Samos, another great Greek astronomer and mathematician, who lived about 250 B.C., established by philosophical deduction and a few simple scientific instruments that the earth revolved around the sun. While Copernicus actually believed himself to be the discoverer of this fact, he but restated the findings advanced by Aristarchus seventeen hundred years earlier.

(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.

(11) To Pythagoras music was one of the dependencies of the divine science of mathematics, and its harmonies were inflexibly controlled by mathematical proportions. The Pythagoreans averred that mathematics demonstrated the exact method by which the good established and maintained its universe. Number therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. After discovering these harmonic ratios, Pythagoras gradually initiated his disciples into this, the supreme arcanum of his Mysteries. He divided the multitudinous parts of creation into a vast number of planes or spheres, to each of which he assigned a tone, a harmonic interval, a number, a name, a color, and a form. He then proceeded to prove the accuracy of his deductions by demonstrating them upon the different planes of intelligence and substance ranging from the most abstract logical premise to the most concrete geometrical solid. From the common agreement of these diversified methods of proof he established the indisputable existence of certain natural laws.

(1) After this we must narrate how, when he had admitted certain persons to be his disciples, he distributed them into different classes according to their respective merits. For it was not fit that all of them should equally participate of the same things, as they were naturally dissimilar; nor was it indeed right that some should participate of all the most honorable auditions, but others of none, or should not at all partake of them. For this would be uncommunicative and unjust. While therefore he imparted a convenient portion of his discourses to each, he benefited as much as possible all of them, and preserved the proportion of justice, by making each a partaker of the auditions according to his desert.

Hence, in conformity to this method, he called some of them Pythagoreans, but others Pythagorists; just as we denominate some men Attics, but others Atticists. Having therefore thus aptly divided their names, some of them he considered to be genuine, but he ordained that others should show themselves to be the emulators of these. He ordered therefore that with the Pythagoreans possessions should be shared in common, and that they should always live together; but that each of the others should possess his own property apart from the rest, and that assembling together in the same place, they should mutually be at leisure for the same pursuits. And thus each of these modes was derived from Pythagoras, and transmitted to his successors.

Again, there were also with the Pythagoreans two forms of philosophy; for there were likewise two genera of those that pursued it, the Acusmatici, and the Mathematici. Of these however the Mathematici are acknowledged to be Pythagoreans by the rest; but the Mathematici do not admit that the Acusmatici are so, or that they derived their instruction from Pythagoras, but from Hippasus. And with respect to Hippasus, some say that he was a Crotonian, but others a Metapontine. But the philosophy of the Acusmatici consists in auditions unaccompanied with demonstrations and a reasoning process; because it merely orders a thing to be done in a certain way, and that they should endeavour to preserve such other things as were said by him, as so many divine dogmas.

They however profess that they will not speak of them, and that they are not to be spoken of; but they conceive those of their sect to be the best furnished with wisdom, who retained what they had heard more than others. But all these auditions are divided into three species. For some of them indeed signify what a thing is; others what it especially is; but others, what ought, or what ought not, to be done. The auditions therefore which signify what a thing is, are such as, What are the islands of the blessed? The sun and moon. What is the oracle at Delphi? The tetractys. What is harmony? That in which the Syrens subsist . But the auditions which signify what a thing especially is, are such as, What is the most just thing?

To sacrifice. What is the wisest thing? Number. But the next to this in wisdom, is that which gives names to things. What is the wisest of the things that are with us, [i. e. which pertain to human concerns]? Medicine. What is the most beautiful? Harmony. What is the most powerful? Mental decision. What is the most excellent? Felicity. What is that which is most truly asserted? That men are depraved. Hence they say that Pythagoras praised the Salaminian poet Hippodomas, because he sings:

(99) "Perfect numbers, therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be. And evil indeed is opposed to evil, but both are opposed to one good. Good, however, is never opposed to good, but to two evils at one and the same time. Thus timidity is opposed to audacity, to both [of] which the want of true courage is common; but both timidity and audacity are opposed to fortitude. Craft also is opposed to fatuity, to both [of] which the want of intellect is common; and both these are opposed to prudence. Thus, too, profusion is opposed to avarice, to both [of] which illiberality is common; and both these are opposed to liberality. And in a similar manner in the other virtues; by all [of] which it is evident that perfect numbers have a great similitude to the virtues. But they also resemble the virtues on another account; for they are rarely found, as being few, and they are generated in a very constant order. On the contrary, an infinite multitude of superabundant and diminished numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite."

(36) The digits 1 and 2 are not considered numbers by the Pythagoreans, because they typify the two supermundane spheres. The Pythagorean numbers, therefore, begin with 3, the triangle, and 4, the square. These added to the 1 and the 2, produce the 10, the great number of all things, the archetype of the universe. The three worlds were called receptacles. The first was the receptacle of principles, the second was the receptacle of intelligences, and the third, or lowest, was the receptacle of quantities.

(60) And the chambers of the winds, and how the winds are divided, and how they are weighed, and (how) the portals of the winds are reckoned, each according to the power of the wind, and the power of the lights of the moon, and according to the power that is fitting: and the divisions of the stars according to their names, and how all the divisions are divided.

(6) And the numbers introduced are sixfold, as three hundred is six times fifty; and tenfold, as three hundred is ten times thirty; and containing one and two-thirds (epidimoiroi), for fifty is one and two-thirds of thirty.

(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

(3) "The Pythagoreans indeed go farther than this, and honour even numbers and geometrical diagrams with the names and titles of the gods. Thus they call the equilateral triangle head-born Minerva and Tritogenia, because it may be equally divided by three perpendiculars drawn from each of the angles. So the unit they term Apollo, as to the number two they have affixed the name of strife and audaciousness, and to that of three, justice. For, as doing an injury is an extreme on the one side, and suffering one is an extreme on the on the one side, and suffering in the middle between them. In like manner the number thirty-six, their Tetractys, or sacred Quaternion, being composed of the first four odd numbers added to the first four even ones, as is commonly reported, is looked upon by them as the most solemn oath they can take, and called Kosmos." (Isis and Osiris.)

(546) the laws which regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and they will bring children into the world when they ought not. Now that which is of divine birth has a period which is contained in a perfect number, 1 but the period of human birth is comprehended in a number in which first increments by involution and evolution [ or squared and cubed] obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. 2 The base of these (3) with a third added (4) when combined with five (20) and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great (400 = 4 × 100), 3 and the other a figure having one side equal to the former, but oblong, 4 consisting of a hundred numbers squared upon rational diameters of a square (i.e. omitting fractions), the side of which is five (7 × 7 = 49 × 100 = 4900), each of them being less by one (than the perfect square which includes the fractions, sc. 50) or less by 5 two perfect squares of irrational diameters (of a square the side of which is five = 50 + 50 = 100); and a hundred cubes of three (27 × 100 = 2700 + 4900 + 400 = 8000). Now this number represents a geometrical figure which has control over the good and evil of births. For when your guardians are ignorant of the law of births, and unite bride and bridegroom out of season, the children will not be goodly or fortunate. And though only the best of them will be appointed by their predecessors, still they will be unworthy to hold their fathers’ places, and when they come into power as guardians, they will soon be found to fail in taking care of us, the Muses, first by under-valuing music; which neglect will soon extend to gymnastic; and hence the young men of your State will be less cultivated. In the succeeding generation rulers will be appointed who have lost the guardian power of testing the metal of your

(91) The oddly-odd, or unevenly-even, numbers are a compromise between the evenly-even and the evenly-odd numbers. Unlike the evenly-even, they cannot be halved back to unity; and unlike the evenly-odd, they are capable of more than one division by halving. The oddly-odd numbers are formed by multiplying the evenly-even numbers above 2 by the odd numbers above one. The odd numbers above one are 3, 5, 7, 9, 11, and so forth. The evenly-even numbers above 2 are 4, 8, 16, 32, 64, and soon. The first odd number of the series (3) multiplied by 4 (the first evenly-even number of the series) gives 12, the first oddly-odd number. By multiplying 5, 7, 9, 11, and so forth, by 4, oddly-odd numbers are found. The other oddly-odd numbers are produced by multiplying 3, 5, 7, 9, 11, and so forth, in turn, by the other evenly-even numbers (8, 16, 32, 64, and so forth). An example of the halving of the oddly-odd number is as follows: 1/2 of 12 = 6; 1/2 of 6 = 3, which cannot be halved further because the Pythagoreans did not divide unity.

(63) Thus thou seest that none of the powers is the first, also none the second, third, fourth or last; but the last generateth the first, as well as the first the last, and the middlemost taketh its original from the last, as also from the first, as well as from the second, third, or any of the rest.

(112) The tetrad--4--was esteemed by the Pythagoreans as the primogenial number, the root of all things, the fountain of Nature and the most perfect number. All tetrads are intellectual; they have an emergent order and encircle the world as the Empyreum passes through it. Why the Pythagoreans expressed God as a tetrad is explained in a sacred discourse ascribed to Pythagoras, wherein God is called the Number of Numbers. This is because the decad, or 10, is composed of 1, 2, 3, and 4. The number 4 is symbolic of God because it is symbolic of the first four numbers. Moreover, the tetrad is the center of the week, being halfway between 1 and 7. The tetrad is also the first geometric solid.

(525) how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply 4 , taking care that one shall continue one and not become lost in fractions. That is very true. Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer? They would answer, as I should conceive, that they were speaking of those numbers which can only be realized in thought. Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth? Yes; that is a marked characteristic of it. And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been. Very true, he said.

(1) Universally, however, it deserves to be known, that Pythagoras discovered many paths of erudition, and that he delivered an appropriate portion of wisdom conformable to the proper nature and power of each; of which the following is the greatest argument. When Abaris, the Scythian, came from the Hyperboreans, unskilled and uninitiated in the Grecian learning, and was then of an advanced age, Pythagoras did not introduce him to erudition through various theorems, but instead of silence, auscultation for so long a time, and other trials, he immediately considered him adapted to be an auditor of his dogmas, and instructed him in the shortest way in his treatise On Nature, and in another treatise On the Gods. For Abaris came from the Hyperboreans, being a priest of the Apollo who is there worshipped, an elderly man, and most wise in sacred concerns; but at that time he was returning from Greece to his own country, in order that he might consecrate to the God in his temple among the Hyperboreans, the gold which he had collected.

Passing therefore through Italy, and seeing Pythagoras, he especially assimilated him to the God of whom he was the priest. And believing that he was no other than the God himself, and that no man resembled him, but that he was truly Apollo, both from the venerable indications which he saw about him, and from those which the priest had known before, he gave Pythagoras a dart which he took with him when he left the temple, as a thing that would be useful to him in the difficulties that would befal him in so long a journey. For he was carried by it, in passing through inaccessible places, such as rivers, lakes, marshes, mountains, and the like, and performed through it, as it is said, lustrations, and expelled pestilence and winds from the cities that requested him to liberate them from these evils.

We are informed, therefore, that Lacedæmon, after having been purified by him, was no longer infested with pestilence, though prior to this it had frequently fallen into this evil, through the baneful nature of the place in which it was built, the mountains of Taygetus producing a suffocating heat, by being situated above the city, in the same manner as Cnossus in Crete. And many other similar particulars are related of the power of Abaris. Pythagoras, however, receiving the dart, and neither being astonished at the novelty of the thing, nor asking the reason why it was given to him, but as if he was in reality a God himself, taking Abaris aside, he showed him his golden thigh, as an indication that he was not [wholly] deceived [in the opinion he had formed of him;] and having enumerated to him the several particulars that were deposited in the temple, he gave him sufficient reason to believe that he had not badly conjectured [in assimilating him to Apollo].

Pythagoras also added, that he came [into the regions of mortality] for the purpose of remedying and benefiting the condition of mankind, and that on this account he had assumed a human form, lest men being disturbed by the novelty of his transcendency, should avoid the discipline which he possessed. He likewise exhorted Abaris to remain in that place, and to unite with him in correcting [the lives and manners] of those with whom they might meet; but to share the gold which he had collected, in common with his associates, who were led by reason to confirm by their deeds the dogma, that the possessions of friends are common . Thus, therefore, Pythagoras unfolded to Abaris, who remained with him, as we have just now said, physiology and theology in a compendious way; and instead of divination by the entrails of beasts, he delivered to him the art of prognosticating through numbers, conceiving that this was purer, more divine, and more adapted to the celestial numbers of the Gods.

He delivered also to Abaris other studies which were adapted to him. That we may return, however, to that for the sake of which the present treatise was written, Pythagoras endeavoured to correct and amend different persons, according to the nature and power of each. All such particulars therefore as these, have neither been transmitted to the knowledge of men, nor is it easy to narrate all that has been transmitted to us concerning him.