The Commentaries on the Timaeus of Plato — trans. Thomas Taylor

Tim, Plat. Vou, I. ot ἜΣ

p. 402

chus, Moderatus, &c.,’ we shall confine ourselves to the three media from which Plato constitutes the soul, arithmetically, geometrically, and harmonically.

It may be seen, however, how all these middles are generated: from equality, by the following method. Phe arithmetic middle, indeed, after this manner: Make the first number equal to the first; the second to the first and the second 3 and the third to the first, second, and third, Three monads therefore being proposed, there will be produced according to this method three terms, vin. 1. 2. 3. preserving an arithmetic middle. Bor this middle consists in equally surpassing according to number, and being equally surpassed.’ But the geometrical middle is produced as follows: Make the first equal to the first; the second, to the first and second ; and the third to the first, to twice the second and the third. For again, there being three monads, there will thus be generated the three terms 1. 2. 4. forming the geometric middle. Por the peculiarity of analogy consists in preserving the same ratio in greater and lesser terms. And the harmonic middle, which has the third order, is generated in the following manner : Three monads being proposed make the first equal to the first, and to twice the second; the second to twiee the first, and twice the second ; and the third τὸ the first, to twice the second, and thrice the third. For by this method the three terms 3. 4. and 6. will be produced, forming the harmonic middle. For the harmonic middle, according to the Platonte definition itself, consists ino surpassing and being surpassed by the same part of the extremes? All the middles, therefore, have their generation from equality.» But if this be the ease, they have the uniform, and a power which collects things, and causes them tobe one. For equality is analogous to sameness, to the monad, to bound and to similitude, through which communion is produced in’ beings. Hence Plato appropriately adds the words, ‘naturally adapted,” beewuse the analogies and all the tiddles have the spontaneous, Por they neither introduce an artificial, nor an adscititious bond, but present themselves to the view in) the essences and powers themselves of things. “For when either in three numbers, or masses, or powers, as is the

* For an account of these media, see my Theoretic Arithmetic. > fu the original, by some negligence of the transertbers, after ww δε urepeyopern, being equally surassed, the words κατὰ ψυσιν αὐτὸν ὁ yidoougys, Lumediately follow, which are obviously totally foreign to this place. > Thus 3 is surpassed by 4, by 1 which is a third part of 3, and 6 surpasses + by 2 which is a third part of 6.

p. 403

> Thus 3 is surpassed by 4, by 1 which is a third part of 3, and 6 surpasses + by 2 which is a third part of 6.

p. 403

middle to the first so is the last to the middle; and again, as is the last to the middle, so is the middle to the first; then the middle becoming both first and fast, and the last and. the first becoming both of them middles, it will thus happen that all of them will necessarily be the same. But becoming the same with each other, they will be one.”

Tn the first place, it is requisite to explain what is here said mathematically ; and in the next place, physically, as being that which is especially proposed to be effected, For it’ is not proper to separate the discussion from. its appropriate theory. There are therefore some who think that Plato in these words detines the geometric middle, and among other things which they assert, they say that the geometrie middle is properly exclusive of all the others analogy ; but that the others may be justly ealled middles. — Nicomachus also is of this opinion, and he speaks rightly. 10) geometric proportion is properly analogy ; but it is requisite to call the ethers middle, as Plato also says further on inthe gencration of the soul, But the others are improperly" called analozics. Vo others, however, these appear not to have apprehended the meaning of Plato properly. For they say that it is not definitely asserted in these words, that there ought to be the same ratio; but thus much only is said, that it is necessary there should be such a habitude of the last to the middle as there is of the middle to the first. But this is common to all the before-mentioned middles, For as the monad is to the duad, according to the arithmetical middle, and the equal in quantity, so is the duad to the triad. | For by as much as the duad is surpassed by the triad, by so much is the triad less than the tetrad. And as the monad is to the duad, according to the geometric middle, so is the duad to the tetrad, For the ratio is the same. And as the triad is to the tetrad, according to the harmonic middle, and the part of the triad by whicn the tetrad surpasses it, so is the tetrad to the hexad. For by that part of the triad by which the triad is exceeded by the tetrad, by the same part of the hexad is the tetrad sarpassed by the hexad, Sach, therefore, is their opinion, though Plato clearly assumes the geometric iniddle. For itis the peculiarity of this proportion, that the first has the same ratio to the middle that the middle has to the third term. As, however, there are three middles, the arithmetic, the geometric, and the harmonic, and these being such as we have shown them to be, Plato very properly assumes these

* For καταχρώμενοι here, 1 read καταχρήστικωε.

p. 404

As, however, there are three middles, the arithmetic, the geometric, and the harmonic, and these being such as we have shown them to be, Plato very properly assumes these

* For καταχρώμενοι here, 1 read καταχρήστικωε.

p. 404

three subjects, numbers, masses, and powers. For the arithmetical middle is in numbers; the geametrical is ina greater degree conversant with continued (than with discrete} quantity ; and the harmonteal middle is in) powers, For it is conversant with sharp and flat sounds, And after this manner you may speak, distinguishing the middles according to their predominance,

All of them, however, may be assumed in numbers, in masses, and Jn powers, And how, indeed, they may be assumed in numbers is evident; for it has been shown by us. But they may also be assumed in masses. For three equal magnitudes being proposed, you may be able, by using the before-mentioned three methods, to devise other magnitudes, at one tine producing an arithmetic, at another a geometric, and at another an harmonic middle. Jn powers likewise after the same manner. For let there be three equal powers, as for instance, three highest Aupateé (vraras) sounds, all of them homotonous, or of the same tone. You will produce therefore from these, the arithmetical middle, if you place the first sound, that is Aypate, equal to the first; but the second, to the first and second, as for instance, another sound emitting a sound the double of the first. And letit be nefe or the last sound, which has a double ratio to Aypate. But the third sound must be placed equal to the first, second, and third. Por it will be a sound which wall have a triple ratio to Aypate, surpassing ucte by as much as γος surpasses Aypate. And these three sounds, Aypate, nete, trite-hyperholéon, Will avithinetically differ from cach other. But you will make the geometrieal middle, if the Aypate being posited, you make the first equal to the first, but the second equal to the first and second hypate. And let this sound be mese : for this is capable of emitting a sound the double of kypate. But if you make the third sound equal to the first Aypate, and to twice the second, and the third, you will have a certain chord which will sound wete-huperbolaon, For this will be capable of producing a sound the double of mese, and the triple of Aypate. These three sounds Likewise will form the geometric middle. ALL the middles, therefore, are seen to exist in numbers, im masses, and in powers. Number, however, is more adapted τὸ the arithmetical, bulk to the geometrical, and power to the harmonic middle. And hence Plato uses these three, viz. numbers, masses, and powers.

It is well, likewise, that assigning certain common ratios, he commences from the middle. For it is this through which all analogy consists, collecting the

———

BOOK 111. TIMASUS OF PLATO. 405