The Obsolesence of Oracles (32-33)

Yes, said I, Theodorus of Soli[*] seems to follow up the subject not ineptly in his explanations of Plato’s mathematical theories. He follows it up in this way: a pyramid, an octahedron, an icosahedron, and a dodecahedron, the primary figures which Plato predicates, are all beautiful because of the symmetries and equalities in their relations, and nothing superior or even like to these[*] has been left for Nature to compose and fit together. It happens, however, that they do not all have one form of construction, nor have they all a similar origin, but the pyramid is the simplest and smallest, while the dodecahedron is the largest and most complicated. Of the remaining two the icosahedron is more than double the octahedron in the number of its triangles. For this reason it is impossible for them all to derive their origin from one and the same matter. For those that are simple and small and more rudimentary in their structure would necessarily be the first to respond to the instigating and formative power, and to be completed and acquire substantiality earlier than those of large parts and many bodies, from which class comes the dodecahedron, which requires

more labour for its construction. Hence it follows that the only primal body is the pyramid, and not one of the others, since by their nature they are outdistanced by it in coming into being. Accordingly, the remedy which exists for this strange state of affairs consists in the division and separation of matter into five worlds, one where the pyramid shall acquire substantiality first, another for the octahedron, and another for the icosahedron; then from the one that first acquires substantiality in each world the rest will have their origin, since a transmutation for everything into everything takes place according to the adaptability of parts to fit together, as Plato[*] himself has indicated, going into the details of nearly all cases. But for us it will suffice to acquire the knowledge in brief form. Since air is formed when fire is extinguished, and when rarefied again gives off fire out of itself, we must observe the behaviour of each of the generative elements and their transmutations. The generative elements of fire are the pyramid,[*] composed of twenty-four primary triangles, and likewise for air the octahedron, composed of forty-eight of the same. Therefore one element of air is produced from two corpuscles of fire combined and united; and that of air again, when divided, is separated into two corpuscles of fire, and again, when compressed and condensed, it goes off into the form of water. The result is that in every case the one which first acquires substantiality always affords the others a ready means of coming into being through transmutation; and it is not one alone that first exists, but another in a different environment is endowed with movement, which takes the lead and forestalls the others in coming into being, and thus the name of being first is kept by all.

Manfully and zealously, said Ammonius, have these matters been worked out by Theodorus; but I should be surprised if it should not appear that he has made use of assumptions which nullify each other. For he insists that all the five shall not undergo construction at the same time, but the simplest always, which requires the least trouble to construct, shall first issue forth into being. Then, as a corollary to this, and not conflicting with it, he lays down the principle that not all matter brings forth the simplest and most rudimentary form first, but that sometimes the ponderous and complex forms, in the time of their coming into being, are earlier in arising out of matter. But apart from this, five bodies having been postulated as primary, and on the strength of this the number of worlds being put as the same, he adduces probability with reference to four only; the cube he has taken off the board, as if he were playing a game with counters, since, because of its nature, it cannot transmute itself into them nor confer upon them the power of transmutation into itself, inasmuch as the triangles are not homologous triangles. For in the others the common triangle which underlies them all is the half-triangle; but in this, and peculiar to it alone, is the isosceles triangle, which makes no convergence towards the other nor any conjunction that would unify the two. If, therefore, there are five bodies and five worlds, and in each one body only has precedence in coming into being, then where the cube has been the first to come

into being, there will be none of the others, since, because of its nature, it cannot transmute itself into any one of them. I leave out of account the fact that they make the element of the dodecahedron, as it is called, something else and not that scalene from which Plato constructs the pyramid and the octahedron and the icosahedron. So, added Ammonius, laughing,either you must solve these problems or else contribute something of your own concerning this difficulty in which we all find ourselves involved.