and "made them as far as possible the fairest and best, out of things which were not fair and
good." The above two sorts of triangles, we are told, are the most beautiful forms, and therefore
God used them in constructing matter. By means of these two triangles, it is possible to
construct four of the five regular solids, and each atom of one of the four elements is a regular
solid. Atoms of earth are cubes; of fire, tetrahedra; of air, octahedra; and of water, icosahedra. (I
shall come to the dodecahedron presently.)
The theory of the regular solids, which is set forth in the thirteenth book of Euclid, was, in
Plato's day, a recent discovery; it was completed by Theaetetus, who appears as a very young
man in the dialogue that bears his name. It was, according to tradition, he who first proved that
there are only five kinds of regular solids, and discovered the octahedron and the icosahedron. *
The regular tetrahedron, octahedron, and icosahedron, have equilateral triangles for their faces;
the dodecahedron has regular pentagons, and cannot therefore be constructed out of Plato's two
triangles. For this reason he does not use it in connection with the four elements.
As for the dodecahedron, Plato says only "there was yet a fifth combination which God used in
the delineation of the universe." This is obscure, and suggests that the universe is a
dodecahedron; but elsewhere it is said to be a sphere. The pentagram has always been
prominent in magic, and apparently owes this position to the Pythagoreans, who called it
"Health" and used it as a symbol of recognition of members of the brotherhood. †It seems that
it owed its properties to the fact that the dodecahedron has pentagons for its faces, and is, in
some sense, a symbol of the universe. This topic is attractive, but it is difficult to ascertain
much that is definite about it.
After a discussion of sensation, Timaeus proceeds to explain the two souls in man, one
immortal, the other mortal, one created by God, the other by the gods. The mortal soul is
"subject to terrible and irresistible affections--first of all, pleasure, the greatest incitement to
evil; then pain, which deters from good; also rashness and fear, two foolish counsellors, anger
hard to be appeased, and hope easily led astray; these they (the gods) mingled with irrational
sense and with all-daring love according to necessary laws, and so framed men."
* See Heath, Greek Mathematics, Vol. I, pp. 159, 162, 294-6.â
€ Heath, loc. cit., p. 161.