(71) Magnitude is divided into two parts--magnitude which is stationary and magnitude which is movable, the stationary pare having priority. Multitude is also divided into two parts, for it is related both to itself and to other things, the first relationship having priority. Pythagoras assigned the science of arithmetic to multitude related to itself, and the art of music to multitude related to other things. Geometry likewise was assigned to stationary magnitude, and spherics (used partly in the sense of astronomy) to movable magnitude. Both multitude and magnitude were circumscribed by the circumference of mind. The atomic theory has proved size to be the result of number, for a mass is made up of minute units though mistaken by the uninformed for a single simple substance.

(1) With respect to your inquiry, “ what the peculiarities are in each of the more excellent genera, by which they are separated from each other? ” if you understand by peculiarities the specific differences under the same genus, which are distinguished by opposite qualities, as the rational and irrational under animal; we by no means admit peculiarities of this kind, in things which neither have one common essence, nor an equal contradistinction, nor receive a composition from something common, which is indefinite, and defines the peculiarity. But if you apprehend the peculiarity to be, as in prior and secondary natures, differing in their whole essence and whole genus, a certain simple condition of being, definite in itself; in this case, your conception of peculiarities will be reasonable. For these peculiarities of things, which have an eternal subsistence, are simple, and entirely exempt. The inquiry, however, proceeds imperfectly. For it was necessary, in the first place, to inquire what the peculiarities are of the more excellent genera, according to essence; in the next place, what they are according to power; and thus afterwards, what they are according to energy.

(43) One of the most surprising features of this discovery is that we finally perceive that the two contrasting sets of qualities are really but two aspects or phases of the whole thing—the real thing, or thing in itself—the unity of the two, instead of being two separated and distinct things. Or, stating it in other words, we discover that the two opposing sets of characteristics are merely relative to each other, and together form a correlated unity and balanced whole.

(11) We divide, therefore, the genus of what is proposed for consideration into the species contained in it; as, in the case of man, we divide animal, which is the genus, into the species that appear in it, the mortal, and the immortal. And thus, by continually dividing those genera that seem to be compound into the simpler species, we arrive at the point which is the subject of investigation, and which is incapable of further division.

(4) All such doubts as these, however, which are adduced foreign to the purpose, and tend from contraries to contraries, we do not consider as pertinent to the present hypothesis. Hence, as we have shown the unappropriateness of them, we do not think it requisite to discuss them any further, because they are contentiously introduced, and not with philosophical investigation.

(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.

(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.

(4) "Dialecticians of the day," replied the Spirit of the River, "all say that the infinitesimally small has no form, and that the infinitesimally great is beyond all measurement. Is that so?" "If we regard greatness as compared with that which is small," said the Spirit of the Ocean, "there is no limit to it; and if we regard smallness as compared with that which is great, it eludes our sight. The infinitesimal is a subdivision of the small; the colossal is an extension of the great. In this sense the two fall into different categories. "Both small and great things must equally possess form. The mind cannot picture to itself a thing without form, nor conceive a form of unlimited dimensions. The greatness of anything may be a topic of discussion, or the smallness of anything may be mentally realized. But that which can be neither a topic of discussion nor be realized mentally, can be neither great nor small.

(24) Now in definitions, difference is assumed, which, in the definition, occupies the place of sign. The faculty of laughing, accordingly, being added to the definition of man, makes the whole - a rational, mortal, terrestrial, walking, laughing animal. For the things added by way of difference to the definition are the signs of the properties of things; but do not show the nature of the things themselves. Now they say that the difference is the assigning of what is peculiar; and as that which has the difference differs from all the rest, that which belongs to it alone, and is predicated conversely of the thing, must in definitions be assumed by the first genus as principal and fundamental.

(69) The Pythagoreans declared arithmetic to be the mother of the mathematical sciences. This is proved by the fact that geometry, music, and astronomy are dependent upon it but it is not dependent upon them. Thus, geometry may be removed but arithmetic will remain; but if arithmetic be removed, geometry is eliminated. In the same manner music depends upon arithmetic, but the elimination of music affects arithmetic only by limiting one of its expressions. The Pythagoreans also demonstrated arithmetic to be prior to astronomy, for the latter is dependent upon both geometry and music. The size, form, and motion of the celestial bodies is determined by the use of geometry; their harmony and rhythm by the use of music. If astronomy be removed, neither geometry nor music is injured; but if geometry and music be eliminated, astronomy is destroyed. The priority of both geometry and music to astronomy is therefore established. Arithmetic, however, is prior to all; it is primary and fundamental.

(74) Number is the term applied to all numerals and their combinations. (A strict interpretation of the term number by certain of the Pythagoreans excludes 1 and 2.) Pythagoras defines number to be the extension and energy of the spermatic reasons contained in the monad. The followers of Hippasus declared number to be the first pattern used by the Demiurgus in the formation of the universe.

(9) the Pure intelligence so called because it purifies the Numerations, it proves and corrects the designing of their representation, and disposes their unity with which they are combined without diminution or division.

(11) For of objects that are moved, some are moved by impulse and appearance, as animals; and some by transposition, as inanimate objects. And of things without life, plants, they say, are moved by transposition in order to growth, if we will concede to them that plants are without life. To stones, then, belongs a permanent state. Plants have a nature; and the irrational animals possess impulse and perception, and likewise the two characteristics already specified.

(2) But it reasonably follows, according to continuity of alliance, that the medium which begins from the most excellent natures, should proceed to such as are less excellent; but that the medium which primarily produces a contact with the last of things, should also in a certain respect communicate with the natures that transcend it. From these media, also, the completion may be seen of the first and last genera, and this entirely connascent, in a similar manner, in existence, in power, and in energy. As we have, therefore, in these two ways, perfectly completed the division of the four genera, we shall deem it sufficient in the others, to exhibit the extreme peculiarities alone, for the sake of conciseness, and because what remains, i. e. the comprehension of the media, is in a certain respect evident. But the media themselves, as being known from the extremes, we shall omit; making a definition of the extremes in the shortest way, as follows.

(3) These divisions are more or less artificial and arbitrary, for the truth is that all of the three divisions are but ascending degrees of the great scale of Life, the lowest point of which is undifferentiated Matter, and the highest point that of Spirit. And, moreover, the different Planes shade into each other, so that no hard and fast division may be made between the higher phenomena of the Physical and the lower of the Mental; or between the higher of the Mental and the lower of the Physical.

(56) Timaeus: when taken singly each in its several kind, is seen by us, but when many are collected together their masses are seen. And, moreover, as regards the numerical proportions which govern their masses and motions and their other qualities, we must conceive that God realized these everywhere with exactness, in so far as the nature of Necessity submitted voluntarily or under persuasion, and thus ordered all in harmonious proportion. From all that we have hitherto said about these Kinds,

(2) But, as your question now stands, with respect to the peculiarities by which these genera are separated, you alone speak of the peculiarities of energies. Hence you inquire concerning the difference in the last things pertaining to them; but you leave uninvestigated such things as are first, and most honourable in them, and which are the elements of their difference. In the same place, also, something is added concerning “ efficacious and passive motions ,” which is a division by no means adapted to the difference of the more excellent genera. For the contrariety of action and passion is not inherent in any one of them; but their energies are unrestrained, immutable, and without habitude to their opposites. Hence, neither must we admit in them motions of such a kind as arise from action and passion. For neither do we admit in the soul a self-motion, which consists of the mover and that which is moved; but we conceive that it is a certain simple essential motion, subsisting from itself, and not possessing a habitude to another thing, and exempt from acting on, and suffering from, itself. Who, therefore, can endure that the peculiarities of the genera superior to the soul, should be distinguished according to active or passive motions?

(533) Dear Glaucon, I said, you will not be able to follow me here, though I would do my best, and you should behold not an image only but the absolute truth, according to my notion. Whether what I told you would or would not have been a reality I cannot venture to say; but you would have seen something like reality; of that I am confident. Doubtless, he replied. But I must also remind you, that the power of dialectic alone can reveal this, and only to one who is a disciple of the previous sciences. Of that assertion you may be as confident as of the last. And assuredly no one will argue that there is any other method of comprehending by any regular process all true existence or of ascertaining what each thing is in its own nature; for the arts in general are concerned with the desires or opinions of men, or are cultivated with a view to production and construction, or for the preservation of such productions and constructions; and as to the mathematical sciences which, as we were saying, have some apprehension of true being—geometry and the like—they only dream about being, but never can they behold the waking reality so long as they leave the hypotheses which they use unexamined, and are unable to give an account of them. For when a man knows not his own first principle, and when the conclusion and intermediate steps are also constructed out of he knows not what, how can he imagine that such a fabric of convention can ever become science? Impossible, he said. Then dialectic, and dialectic alone, goes directly to the first principle and is the only science which does away with hypotheses in order to make her ground secure; the eye of

(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

(1) But neither must we admit that cause of the distinction of these genera which you subjoin, viz. “ that it is an arrangement with reference to different bodies; as, for instance, of Gods to etherial bodies, but of dæmons to aerial bodies, and of souls to such as are terrene .” For such an arrangement as this, which resembles that of Socrates to a tribe, when he is a senator, is unworthy of the divine genera, because all of them are essentially unrestrained and free. To which may be added, that it is dreadfully absurd to ascribe to bodies a principal power of giving a specific distinction to the first causes of themselves. For bodies are in servile subjection to these causes, and are ministrant to generation. And farther still, the genera of the more excellent natures are not in bodies, but the former externally rule over the latter. Hence they are not changed in conjunction with bodies. Again, they impart from themselves to bodies every such good as they are able to receive, but they themselves receive nothing from bodies; so that neither will they derive from them certain peculiarities.