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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.
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