here are any distinct kinds of visible bodies fairer than these.
Wherefore we must endeavour to construct the four forms of bodies
which excel in beauty, and then we shall be able to say that we have
sufficiently apprehended their nature. Now of the two triangles,
the isosceles has one form only; the scalene or unequal-sided has
an infinite number. Of the infinite forms we must select the most
beautiful, if we are to proceed in due order, and any one who can
point out a more beautiful form than ours for the construction of these
bodies, shall carry off the palm, not as an enemy, but as a friend.
Now, the one which we maintain to be the most beautiful of all the many
triangles (and we need not speak of the others) is that of which the
double forms a third triangle which is equilateral; the reason of this
would be long to tell; he who disproves what we are saying, and shows
that we are mistaken, may claim a friendly victory. Then let us choose
two triangles, out of which fire and the other elements have been
constructed, one isosceles, the other having the square of the longer
side equal to three times the square of the lesser side.

Now is the time to explain what was before obscurely said: there was an
error in imagining that all the four elements might be generated by and
into one another; this, I say, was an erroneous supposition, for
there are generated from the triangles which we have selected four
kinds--three from the one which has the sides unequal; the fourth
alone is framed out of the isosceles triangle. Hence they cannot all be
resolved into one another, a great number of small bodies being combined
into a few large ones, or the converse. But three of them can be thus
resolved and compounded, for they all spring from one, and when the
greater bodies are broken up, many small bodies will spring up out
of them and take their own proper figures; or, again, when many small
bodies are dissolved into their triangles, if they become one, they will
form one large mass of another kind. So much for their passage into one
another. I have now to speak of their several kinds, and show out of
what combinations of numbers each of them was formed. The first will be
the simplest and smallest construction, and its element is that triangle
which has its hypotenuse twice the lesser side. When two such triangles
are joined at the diagonal, and this is repeated three times, and the
triangles rest their diagonals and shorter sides on