In the first place, then, as is evident to all, fire and earth
and water and air are bodies. And every sort of body possesses
solidity, and every solid must necessarily be contained in
planes; and every plane rectilinear figure is composed of
triangles; and all triangles are originally of two kinds, both of
which are made up of one right and two acute angles; one of them
has at either end of the base the half of a divided right angle,
having equal sides, while in the other the right angle is divided
into unequal parts, having unequal sides. These, then, proceeding
by a combination of probability with demonstration, we assume to
be the original elements of fire and the other bodies; but the
principles which are prior to these God only knows, and he of men
who is the friend of God. And next we have to determine what are
the four most beautiful bodies which are unlike one another, and
of which some are capable of resolution into one another; for
having discovered thus much, we shall know the true origin of
earth and fire and of the proportionate and intermediate
elements. And then we shall not be willing to allow that there
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.