Greek Texts & Translations

Moreover, a point about which our previous statement was obscure must now be defined more clearly. It appeared as if the four Kinds, 54cin being generated, all passed through one another into one another, but this appearance was deceptive. For out of the triangles which we have selected four Kinds are generated, three of them out of that one triangle which has its sides unequal, and the fourth Kind alone composed of the isosceles triangle. Consequently, they are not all capable of being dissolved into one another so as to form a few large bodies composed of many small ones, or the converse; but three of them do admit of this process. For these three are all naturally compounded of one triangle, so that when the larger bodies are dissolved many small ones will form themselves from these same bodies, receiving the shapes that befit them; 54dand conversely, when many small bodies are resolved into their triangles they will produce, when unified, one single large mass of another Kind. So let thus much be declared concerning their generation into one another.

In the next place we have to explain the form in which each Kind has come to exist and the numbers from which it is compounded. First will come that form which is primary and has the smallest components, and the element thereof is that triangle which has its hypotenuse twice as long as its lesser side. And when a pair of such triangles are joined along the line of the hypotenuse, and this is done thrice, by drawing the hypotenuses 54eand the short sides together as to a center, there is produced from those triangles, six in number, one equilateral triangle. ^note And when four equilateral triangles are combined so that three plane angles 55ameet in a point, they form one solid angle, which comes next in order to the most obtuse of the plane angles. And when four such angles are produced, the first solid figure ^note is constructed, which divides the whole of the circumscribed sphere into equal and similar parts. And the second solid ^note is formed from the same triangles, but constructed out of eight equilateral triangles, which produce one solid angle out of four planes; and when six such solid angles have been formed, the second body in turn is completed. 55bAnd the third solid ^note is composed of twice sixty of the elemental triangles conjoined, and of twelve solid angles, each contained by five plane equilateral triangles, and it has, by its production, twenty equilateral triangular bases.

Now the first of the elemental triangles ceased acting when it had generated these three solids, the substance of the fourth Kind ^note being generated by the isosceles triangle. Four of these combined, with their right angles drawn together to the center, produced one equilateral quadrangle; and six such quadrangles, 55cwhen joined together, formed eight solid angles, each composed of three plane right angles; and the shape of the body thus constructed was cubic, having six plane equilateral quadrangular bases. And seeing that there still remained one other compound figure, the fifth, ^note God used it up for the Universe in his decoration thereof.

Now in reasoning about all these things, a man might question whether he ought to affirm the existence of an infinite diversity of Universes or a limited number; and if he questioned aright he would conclude that the doctrine of an infinite diversity is that of a man unversed ^note 55din matters wherein he ought to be versed; but the question whether they ought really to be described as one Universe or five is one which might with more reason give us pause. Now our view declares the Universe to be essentially one, in accordance with the probable account; but another man, considering other facts, will hold a different opinion. Him, however, we must let pass. But as for the Kinds which have now been generated by our argument, let us assign them severally to fire and earth and water and air. To earth let us give the cubic form; for of the four Kinds earth is the most immobile 55eand the most plastic body, and of necessity the body which has the most stable bases must be pre-eminently of this character. Now of the triangles we originally assumed, the basis formed by equal sides is of its nature more stable than that formed by unequal sides; and of the plane surfaces which are compounded of these several triangles, the equilateral quadrangle, both in its parts and as a whole, has a more stable base than the equilateral triangle.