To set out the sides of the five figures and to compare them with one another; and that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another.
Proof
Compare the side lengths: tetrahedron 2/3, octahedron
1/2, cube 1/3, icosahedron (minor irrational),
dodecahedron (apotome). For the uniqueness clause: at each vertex of
a regular polyhedron, the sum of face-angles must be less than four
right angles (XI.21). Equilateral triangles (60∘): 3, 4, or
5 around a vertex — tetrahedron, octahedron, icosahedron. Squares
(90∘): only 3 around a vertex — cube. Regular pentagons
(108∘): only 3 around a vertex — dodecahedron. Hexagons
(120∘): three would tile flat, no vertex — impossible.
Larger polygons: even three exceed 360∘. Hence exactly five
regular polyhedra exist.