{(3) The construction of the regular solids. 
Plato, of course, constructs the regular solids by simply 
putting together the plane faces. These faces are, he observes, 
made up of triangles; and all triangles are decomposable into 
two right-angled triangles. Right-angled triangles are either 
(1) isosceles or (2) not isosceles, having the two acute angles 
unequal. Of the latter class, which is unlimited in number, 
one triangle is the most beautiful, that in which the square on 
the perpendicular is triple of the square on the base (i. e, the 
triangle which is the half of an equilateral triangle obtained 
by drawing a perpendicular from a vertex on the opposite 
side). (Plato is here Pythagorizing. 1 ) One of the regular 
solids, the cube, has its faces (squares) made up of the first 
kind of right-angled triangle, the isosceles, four of 
them being put together to form the square; three 
others with equilateral triangles for faces, the tetra 
hedron, octahedron and icosahedron, depend upon 
the other species of right-angled triangle only, 
each face being made up of six (not two) of those right-angled 
triangles, as shown in the figure; the fifth solid, the dodeca 
hedron, with twelve regular pentagons for 
faces, is merely alluded to, not described, in 
the passage before us, and Plato is aware that 
its faces cannot be constructed out of the two 
elementary right-angled triangles on which the 
four other solids depend. That an attempt was made to divide 
the pentagon into a number of triangular elements is clear 
from three passages, two in Plutarch 2 
and one in Alcinous. 3 Plutarch says 
that each of the twelve faces of a 
dodecahedron is made up of thirty 
elementary scalene triangles which are 
different from the elementary triangle 
of the solids with triangular faces. 
Alcinous speaks of the 360 elements 
which are produced when each pen 
tagon is divided into five isosceles triangles and each of the