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

3 For riiOus here, read Au os.

* For 12 contains 8 once, and the half of 8. And ina similar manner 4%=1} and 77=11),

δ Fos 12=2x 2X3, 2 being the cube root of 8, and 3 being the cube root of 27; and 18=3X3x 2.

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as in the media of the before-inentioned cubes. Hence between similar solids, two media are sufficient; just as Plato says, that two media adapt solids to each other, but never one medium. What then, some one may say, is there not one medium alone of the two solid numbers 64 and 729, which medium is 216? For G4 is a cube produced from 4, but 729 from 9. And 729 is the triple and superparticular ogdoan part of 216; and after the same manner 216 of 61. For each contains the other thrice, and three eighths of it besides.". And this will not only be the case in these, but also in other numbers: for these are the smallest numbers which admit of this. In answer to this however it must be said, that the above-mentioned numbers are cubes and at the same time squares; the one, i. e. 64, being the square of 8, but the other, i. 6. 729, being the square of 27. Hence they have one mean, not so far as they are cubes, but so far as they have the tetragonic peculiarity, For the tetragonic side of 64, i. e. 8, being multiplied by 27, which is the tetragonic side of 729, produces the analogous mean 216, according to the method delivered [by mathematicians] of finding the mean between two squares, He who makes the objection, therefore, using solids not as solids, binds them together by one medium. But if he had surveyed them so far as they are solid numbers and cubes, he would have found that there are also two media between these, the one being 144, from four times 4 multiplied by 9, but the other 324, from nine times 9 multiplied by 4.

But Democritus + doubts, how it is said that one analogous medium falls between two planes. Vor by assuming four lines in continued proportion, it may be shown that the squares from them are analogous;’ so that two analogous media will fall between two extreme planes. He adds, that different persons have been involved in different difficulties through this doubt, and have been led by it to the duplication of the cube, and such-like investigations. Plato, however, does not say that one medium only falls between any casual planes, nor again two media between casual solids, but between those that are similar, and in an eflable ratio, and which have their sides arranged according to numbers. For

* Thus 729 contains three times 216, i. ὁ. 648, and three eighths of it besides. For the eighth part of 216 is 27, and thrice 27 is 81, the difference between 729 and 648. And thus also 216 contains 64 thrice, i, e. 192, and 24 besides, which is three cighths of 64. ᾿

* This is most probably the junior Democritus mentioned by Porphyry in his life of Plotinus,

For the eighth part of 216 is 27, and thrice 27 is 81, the difference between 729 and 648. And thus also 216 contains 64 thrice, i, e. 192, and 24 besides, which is three cighths of 64. ᾿

* This is most probably the junior Democritus mentioned by Porphyry in his life of Plotinus,

" For let the lines be as the numbers 2. 4. 8. 16, which are in continued proportion; then the squares of them 4, 16. 64. 256 will also be analogous. Foras 4is to 10, so is 16 to 64, and so is

64 to 256.

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the things generated by the demiurgic God, are effable with reference to each other, and are variegated by demiurgic numbers, as Plato says in another part of this dialogue. And it is requisite to assume similar planes, and solid numbers, and to survey in these the truth of the Platonic assertion. We shall show therefore at the end of these Commentaries, how itis possible, two right lines being civen, to find two analogous media, selecting for this purpose the demonstration of Archytas, rather than that of Menwchmus, because he uses conical lines, and in like manner rather than that of Eratosthenes, because he employs the apposition of a rule."

With respect however to the things investigated, it must now be said that Plato appears to have perfect contidence in arithmetical demonstrations, since it is also possible to find in geometrical figures of two solids an analogous medium. For if there are three analogous right lines, a, 2, y, in a duple ratio, the squares from them will be in a quadruple ratio, as £4, ior, and ὃ. But the solids from them will bein an octuple ratio, as 913, 25, and y. Hence there will be three cubes, the extremes of which will have one analogous mean, And it is manifest that all cubes are similar to each other. For the angles are the same in each cube and are equal 5 and they are also comprehended by similar planes ; and the multitude of them is equal. Moreover, we may thus demonstrate in the same manner as Democritus, that two analogous media fall between two similar solids. For that all squares are similar to each other is evident ; since the angles are the same in each, and are equal, and the sides are analégous. Hence it seems that Plato employing numbers, shows that solids are never co-adapted by one mean, but always by two media. For in these, as you see, the extremes are cubes, and at the same time similar planes. For 613, is from usr, and az. But the other of the extremes 7, is from Β, and 4, and there is the same ratio of the sides. There is therefore one medium of these, so far as they are similar planes, but not so far as they are solids. So that you will have the solution of what is said, by assuming numbers. Tor it is possible to find the same numbers which are at the same time similar solids and planes; but it isimpossible to assume geometrical figures which are at one and the same time similar planes and solids; since this also may be said, that all of them being cubes, the form of them is one. But Plato

* From the most unfortunate loss of the latter part of these commentaries of Proclus, this method likewise of his fiuding two analogous media between two given lives, is lost.

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