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

If, however, there is only one element of the world, the variety of the phenomena, the opposition of the cir-

» Por δεειδεν here, read &voedkes. > Ove is omitted in this place in the original.

2 Ory is also here omitted in the original.

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culations, and the war of generation, will be subverted, and either all things will be perpetual, or all things will be corruptible. Butif there is not one element only, there will either be two elements, or more than two. And if two, they will either be contraries, or not contraries. Tf, however, they are not contraries, there will neither be action, nor passion, nor opposition in bodies, nor will there be genera. tion in things which have generation. But if they are contraries, these will require amedinm. And if this be the case, there will either be one medium, or two media, Itis impossible, however, that there should be only one medium: for the elements are not superticies. Hence there are two media. But if there are two media of two things, all are four, That so many elements therefore in number are suflicient to the world, is through these things manifest.

Let us, however, if you please, concisely survey the mathematical meaning of the words before us, and afterwards adduce the physical theory pertaining to them. For how of two similar superticies or planes there is one medium, and of two similar solids two media, we will survey in number by themselves. For the primordial and spontaneous nature of numbers, is to be embraced prior to geometrical necessity. Tn the first place, therefore, let there be two square numbers 9 and 16, the less of which has for its side 3, and the greater 4, By multiplying these and making 12, we shall have an analogy in the three terms 9, 12, and 16." Let two numbers likewise be assumed, which are not squares indeed, but at the same time are similar planes, and Jet them be 18 and 3.2," the former being generated from the triad and hexad, but the latter from the tetrad and ogdoad. [{ therefore, we multiply either the triad by the ogdoad, or the hexad by the tetrad, we shall have for the product 24, binding in analogy 18 to 32, according to a sesquitertian ratio. This, however, is caused by their sides having the same ratio, If, therefore, the sides of the assumed numbers are found to receive no analogous mean or medium, all the planes generated from them will have but one medium, according to the before-mentioned mode,’ But if the sides themselves should be

* For as 9 isto 12, sois 12to 10. Ὁ The two similar plane numbers 18 and 32 here adduced by Proclus, prove that Gaston Pardies was greatly mistaken in asserting in his Elements of Geometry, “ that if two numbers are similer planes,

Ὁ The two similar plane numbers 18 and 32 here adduced by Proclus, prove that Gaston Pardies was greatly mistaken in asserting in his Elements of Geometry, “ that if two numbers are similer planes, the greater may be divided into as many squares as there are units in the less.” See the Translation of this work by Harris, p. 133. For 32 eannot be divided into as many squares as there are uoits in 18. And 32 and 18 are evidently similar plane vumbers, because their sides are analogous. For as 3 is to 6 so is 4 to 8. Υ

2 The sides of these numbers are 3, 6, 4, and 8, and they have no analogous mean. For there is no geometrical mean between 3 and 6, nor between 4 and 8, [lence the planes generated from them, vis.

18 and 32, will have but one medium, which is 24,p. 412 found to receive a certain analogous mean, the planes also produced from them, will necessarily receive more than one mean. For let there be two squares 16 and 81, and let the side of the former be 4, but of the latter 9. Since, therefore, the analogous medium between 4 and 9 is 6, according to a sesquialter ratio, it is necessary that more than one mean should fall between them. For the tetrad multiplied by the hexad will produce 24; but the hexad multiplied by itself will produce the square of itself 36; and multiplied by 9, will produce 564. And there will be a continued analogy, in the terms 16, 24, 36, 54, 81." Hence, when the sides have an analogous mean, the planes produced from them, will have more than one mean. Hence, too, Plato appears to me to say very cautiously, not that there is entirely one medium in similar planes, but that it is possible for one to be suflicient. For more than one plane being produced, one medium would be sufficient to conjoin them, viz. 36 alone, according to the duple sesquiquartan ratio.” And thus much concerning similar planes.

For more than one plane being produced, one medium would be sufficient to conjoin them, viz. 36 alone, according to the duple sesquiquartan ratio.” And thus much concerning similar planes.

Let us, however, now pass onto similar solids, and survey the media in these. In the first place, therefore, let there be two cubes 8 and 27, the former having for its side 2, and the latter 8. Of these cubes, there will be two media, the one being produced from twice two multiplied by three, 1. e. 12, and which on this account 15 (4exis) a Seam, but the other from thrice three multiplied by two, 1. e. 18, and which is therefore (πλίνθος ᾽) a tile. These will make a continued analogy with the before-mentioned cubes, according to ἃ sesquialter ratio.” And here you taay see how each of the media has two sides from the cube placed next to it, but the remaining side from the other cubes This however will be useful to us for the purposes of physiology. Again, ifthe numbers were not cubes, but similar solids, they will likewise have two analogous middles or means. For let there be two similar solids 24 and 192, the sides of the former being 2, 3, 4, but of the latter 4, 6, 8. And from the duad, the triad, and the ogdoad, 48 will be produced, but from the tetrad, the hexad, and again the tetrad, the product will be 96. Ilere too, each of the media will have two sides from that similar solid of the extremes which is next to it, but one side fromthe other cube, in the same manner

' For as 16 is to 24, sois 24 to 36, sois 36 to 54, so is 54 to 81, the ratio being sesquialter.

* For 36 is a geometrical mean between 16 and 81, according to a duple sesquiquartan ratio, For 36 contains 16 twice, and a fourth part of it, i.e. 4also; aud 81 contains 36 twice, or 72, and 0 besides, which is a fourth part of 36.

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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