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 the same
point as a centre, a single equilateral triangle is formed out of
six triangles; and four equilateral triangles, if put together,
make out of every three plane angles one solid angle, being that
which is nearest to the most obtuse of plane angles; and out of
the combination of these four angles arises the first solid form
which distributes into equal and similar parts the whole circle
in which it is inscribed. The second species of solid is formed
out of the same triangles, which unite as eight equilateral
triangles and form one solid angle out of four plane angles, and
out of six such angles the second body is completed. And the
third body is made up of 120 triangular elements, forming twelve
solid angles, each of them included in five plane equilateral
triangles, having altogether twenty bases, each of which is an
equilateral triangle. The one element (that is, the triangle
which has its hypotenuse twice the lesser side) having generated
these figures, generated no more; but the isosceles triangle
produced the fourth elementary figure, which is compounded of
four such triangles, joining their right angles in a centre, and
forming one equilateral quadrangle. Six of these united form
eight solid angles, each of which is made by the combination of
three plane right angles; the figure of the body thus composed is
a cube, having six plane quadrangular equilateral bases. There