But through this harmony and analogy, in the first place, sameness presents itself to the view, and in the next place union. For bodies themselves according to their own nature are partible, aud are subdued by differences and strife. “These, however, at the same time through harmony, are leagued in tdendship with same- ness, and through sameness with union. For through analogy the universe 15
' Por ev avry read ey avros.
BOOK 111. TIM.£Us OF PLATO. 409
completely rendered one, this having the power of making things that are divided to be one, of congregating things that are multiplied, and connecting things that are dissipated. Hence, theologists surveying the causes of these things in the Gods, enclose Venus with Mars, and surround them with Vulcanian bonds; the difference which isin the world being connected through harmony and friendship. All this complication and connexion likewise has Vulcan for its cause, who through demiurgic bonds connects sameness with difference, harmony with dis- cord, and communion with contrariety. And this being eflected, Apollo, Her- mes, and each of the Gods laugh. But their laughter gives subsistence to mun- dane natures, and inserts etlicacious power in the bonds. Let these things, how- ever, as it is said, be preserved in sacred silence. But now, from what has been discussed, let thus much be manifest to us, that the physical bond being Vulcanian and demiurgic, (for the one and all-perfect Demiurgus comprehends also the pro- duction which is through necessity, as being Vulcanian and Dionysiacal, and causing each of the parts of the universe to be a whole,) is collective of contraries, and connective of material things; uniting their essences, measuring their masses, and harmonizing their powers. It likewise makes all things to be in all, and exhibits the same things in each other, according to all possible modes, empy- really, aerially, aquatically, and terrestrially.
“ Tf then it were necessary that the body of the universe should have been generated a superficies, and not have depth, one medium might have been suticient for the purpose of binding both the natures that sub- sist with it, and itself. But now it is requisite that it should be a solid, and solids are never adapted to each other by one, but always by two
media.”
The scope proposed to us (in the Timeus], is, as we have before observed, to learn how the universe is constituted, and of what it consists. But this being the design, we may see in what a well-ordered manner the discourse devises. the composition of the four elements. For it is impossible that there should be one simple element alone; since there would not be generation. For all generation is ascertain mutation. But all mutation is naturally adapted to be effected in two things. All generation likewise is from contraries. But a simple element itself, is by no means contrary to itself: for it would be itself corruptive of itself. If, therefore, it is necessary there should be generation, it is
Tim. Plat. Ὁ θῖν ἂν ᾿ 3F
410 PROCLUS ON THE [BOOK 115.
necessary there should not be one element only. For as Hippocrates says, if there was one element only, it would be impossible for things to be changed. For mutation and motion are not to the similar, but to the contrary. Hence there is not one simple element only. If, however, there is not one, but two at least, it is necessary that these should be contraries: for generation 15 from contraries. It is necessary, therefore, that there should be two elements having in a becoming manner a nature contrary to each other. Hence, if they are contraries, they will be in want of a certain bond and medium, For it is impossible that two contraries can ina becoming manner coalesce, without a third thing ; since it is necessary that a bond should intervene, which is collective of both. For being themselves contraries, they will avoid communion with each other. Hence it is necessary, there should be another third thing which conjoins them and leads them to the completion ofone thing, But it is likewise necessary that this medium should be of a biformed' nature. For if the elements which were to be bound were superticies, one medium would be sufficient, But since they are solids, they are connected through two media. For the duad being the primary leader of solids, is also allotted the primordial cause of the bonds that are in them. Hence, likewise, Timieus calls a binding of this kind harmony, as inserting in the extremes ἃ symmetry of communion with each other. The analogy also which is in solids is introduced through two media. For two media analogously come between two similar solids. If, therefore, these things are rizhtly asserted, all the elements are four; and there is neither one alone, lest we should destroy mutation; nor two contraries without a third thing, lest there should not* be a bond of things which are hostile to each other, Por there will not be order and ornament from two things most foreign to each other, Butif you con- ceive a certain thing of this kind, the solution of the doubt will be easy, Moreover, neither will there be alone two things which are not! contravies. For they will not be able to operate on each other, Por whiteness suffers nothing from ἃ line, but from blackness. Nor does heat suffer any thing from whiteness, but from cold,
Again, therefore, it must be said still more universally, reasoning from things known, that either there is one clement alone, or not one. 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.
Book 111.} Tim.Eus of Plato. 411
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 geome- trical 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 gene- rated 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,
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,
412 PROCLUS ON THE [BooR 11}.
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 suf- ficient 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 ex- tremes 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),