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 su- perparticular 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 be- tween 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 be- tween 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 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.
414 PROCLUS ON THE (BooK 111.
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 there- fore 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 apposi- tion 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 assum- ing 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.
poor 11 TIM.EUS OF PLATO. 415
assuming that the means are similar to the extremes, is thus confident in the theorem. For how would the extremes be in want of other bonds, if they had entirely the same form? And how would the media communicate with the extremes, and differ from them according to the sides, if they were all of them cubes? Hence it is evident that he assumes the media, as being truly media, and thus says, that solids are never conjoined by one, but always by two media ; every mediam containing the communion and difference of the things of which it is the medium. For to say universally, that all solids are connected through two media, makes the media to be infinite. It is manifest, therefore, that he assumes things which are most distant, and in every respect contrary to each other, and which have all the sides opposite to all; these in natural bodies being corporeal powers. But le does this, in order that of the media, one of them may have a greater communion with one of the extremes, but a less with the other; and that this may be vice versa with the other medium. Unless that also is true which is asserted by our preceptor. For he says that it is necessary to assume the same ratio in the media or means, as there is in the sides of the extremes. Thus, for instance, if one of the cubes is 8, but the other 27, we shall find the media of them, if we take their sides 2 and 3, and multiply the square of 2, i. e. 4 by 3, and the square of 3,1. 6. 9 by 2, For then the media will be 12, and 18, which will conjoin the extremes through the sesquialter ratio, which is the ratio of the sides of the cubes. Hence, as there is the same ratio in the sides of the cubes, and in the media, Plato says that there is necessarily two media, and this in amanner more consonant to the proposed physiology. For in the powers of the elements, and in simple forms, the Demiurgus inserted communion prior to
_ things of a composite nature. We however conjoin the extremes through the
octuple ratio, the sides of them not having an octuple ratio. For the mean being assumed in a duple ratio, the extremes will have a quadruple ratio. Thus, for instance, in the three proposed terms, if we assume a fourth analogous term, we shall find that as the side 2 is octuple of the side 16, so the first cube is con- joined to the fourth through the octuple ratio. For if you add 16, as a fourth term to 2. 4. 8, the cube from 16, is conjoined to 8, through the octuple ratio which 64 has to 8, and 512 to 64, and 4096 which is the cube of 16;to 512." So that the sides of the media receiving an octuple ratio, two media will fall between
" If each of the terms 2. 4. 8. 16, is cubed, the four terms 8. 64. 512. 4096, will be produced, and C4 is octuple of 8, 512 of 64, and 4096 of 512; and the first cube 8 is conjoined to the fourth 4096, through the octuple ratio.
416 PROCLUS ON THE [Boox 111.
the extremes. But ifa fifth analogous term is added, the sides will no longer be conjoined in an octuple ratio but in the ratio of 16 to 15 and on this account there will be three analogous media between the two cubes. What Plato says, therefore, is true according to the before-mentioned method. Are not the sides also co- adapted to his purpose? And it is requisite to say, that there may be one medium between two cubes, yet not according to harmonic ratios. Hence, when there is truly a colligation of the extremes through these ratios, then it is perfectly requisite that there should be two media. Through these things, therefore, it is manifest mathematically, that similar planes require one medium, and similar solids two media, and that they can never be bound by one medium alone.
Being impelled, however, by these observations, let us sce how physical concep- tions accord with them, and let us adapt probable to scientific assertions. And in the first place, let us survey whata physical plane is, and how in planes of this kind, there is one medium, but two in physical solids. The divine Tamblichus indeed (for this man in a remarkable degree comprehended a theory of this kind, others being as it were asleep, and conversant only with the mathematical meaning of the words) appears to me to distinguish things simple from such as are composite, parts from wholes, and in short, material powers, and material forms, from the essences to which they give completion. And some of these, he calls superficies, but others solids. For as a superticies is the ultimate boundary of a mathe matical body, so likewise material form, and material power, are the morphe and boundary of their subjects. ‘These things, therefore, being thus divided, in things of a simple nature, one medium is suflicient, because there is [one] difference of the reasons and forms, and according to the common bonds of the reasons and the life. Lorin these there is one medium, Hence quality 15. uniformly connected with quality, and power with power, according to the difference and sameness of forms. But in things of a composite nature, there are very properly two media. For the duad is the supplier of all composition and separation. Every composite nature however consists of many essences and powers. Hence, there are many media. And these at least are two-fold. For there is one medium according to form, and another according to subject.
We however, conformably to physical principles, speak as follows, receiving auxiliaries from what Plato says as he proceeds. Or rather, let us speak from the beginning. There are some physiologists then, who ascribe one power to each of the elements ; to fire indeed heat, to air frigidity, to water moisture, and
’ ty earth dryness; in so doing entirely wandering from the truth. In the first