Stephanus 54

54a Timaeus: Of the two triangles, the isosceles has taken one nature; the oblong one, kinds without limit. So we must choose, in turn, the most beautiful of the unlimited ones, if we are to make a beginning in the right way. If anyone has a finer choice to name for their construction, he—as a friend, not an enemy—wins the day; but we take as the most beautiful of the many triangles one alone, passing over the rest: the one out of which the equilateral triangle was put together in thirds. 54b Why this is so would take a longer account; but for whoever tests it and finds it to be indeed so, friendly prizes lie laid up. Let it then be settled that two triangles were chosen, out of which the body of fire and the bodies of the rest were devised: the one isosceles, the other always having the greater of its sides triple in square of the lesser. What was said before obscurely must now be marked off more precisely. The four kinds, then, seemed all to have their coming-to-be through one another—an appearance not correct: for the four kinds come to be from 54c the triangles we chose, three of them from the one that has unequal sides, while the fourth alone was fitted together from the isosceles triangle. So it is not possible for all of them, dissolving into one another, to become few and large from many and small, and the reverse; but the three can. For since all of them are by nature from one triangle, when the larger are dissolved, many small ones will be formed out of those same triangles, taking on the shapes proper to them; and when small ones, again, are many and scattered into 54d the triangles, they become one number of one mass and can complete one other large kind. Let this much, then, have been said about their generation into one another; but of what sort each came to be in shape, and out of how many numbers falling together, it would be next in order to say. The beginning will be the first and smallest form constructed; its element is the triangle that has its hypotenuse double the shorter side in length. And when two such are joined along the diagonal, and this is done 54e three times, the diagonals and the short sides being brought to rest on the same point as center, one equilateral triangle has come to be out of six in number.

Timaeus: and equilateral triangles, when four come together with every three plane angles, make one solid

54a Timaeus: Of the two triangles, the isosceles has one nature, while the unequal-sided one has infinitely many. So we must pick out the most beautiful of the infinite set, if we are to make a proper start. If someone can pick out a more beautiful one and name it for their construction, he wins—he being not an enemy but a friend. We, then, pick one as the most beautiful of the many triangles and pass over the rest: the one from which the equilateral triangle is put together out of three. 54b The reason for this would take a longer account. But for anyone who tests this and finds that it is so, the prizes are laid out in friendship. So let two triangles be picked out, the ones from which fire and the other bodies have been constructed: the isosceles, and the one whose longer side always has three times the power of the shorter side. What was said unclearly before must now be marked out more clearly. For it seemed that all four kinds came to be from one another and into one another, but that was an incorrect impression. Four kinds do come to be from 54c the triangles we have picked: three of them from the single one that has unequal sides, and the fourth fitted together from the isosceles triangle alone. So not all of them can break apart into one another, so that a few large ones come to be out of many small ones, or the other way around; but the three can. Since all of them grow from one, when the larger ones are broken up, many small ones will form out of the same parts, taking on the shapes that belong to them. And when many small ones are scattered apart along their triangles, 54d one number of them, forming a single mass, would produce one other single form, a large one. Let this much be said about their coming to be from one another. Now, in order, we should tell what sort of form each of them has come to be, and out of what numbers of parts put together. The first and smallest form to be put together will lead off, and its element is the triangle whose hypotenuse is twice the length of its shorter side. When two of these are put together along their diagonals, and this is done three times, 54e with the diagonals and the short sides all brought to meet at a single center, one equilateral triangle is formed out of six of them.

Timaeus: And when four equilateral triangles are put together, joining three plane angles into one solid

54a their nature adequately.

Timaeus: Now of the two triangles, the isosceles possesses one single nature, but the scalene an infinite number; and of these infinite natures we must select the fairest, if we mean to make a suitable beginning. If, then, anyone can claim that he has chosen one that is fairer for the construction of these bodies, he, as friend rather than foe, is the victor. We, however, shall pass over all the rest and postulate as the fairest of the triangles that triangle out of which, when two are conjoined, 54b the equilateral triangle is constructed as a third. The reason why is a longer story; but should anyone refute us and discover that it is not so, we begrudge him not the prize. Accordingly, let these two triangles be selected as those wherefrom are contrived the bodies of fire and of the other elements,— one being the isosceles, and the other that which always has the square on its greater side three times the square on the lesser side.

Moreover, a point about which our previous statement was obscure must now be defined more clearly. It appeared as if the four Kinds, 54c in 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; 54d and 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 54e and the short sides together as to a center, there is produced from those triangles, six in number, one equilateral triangle.

And when four equilateral triangles are combined so that three plane angles