(Greek) = 'numbers squared from' or 'upon diameters'; (Greek) = 'rational,' i.e.omitting fractions, (Greek), 'irrational,' i.e.
including fractions; e.g.49 is a square of the rational diameter of a figure the side of which = 5: 50, of an irrational diameter of the same.
For several of the explanations here given and for a good deal besides I am indebted to an excellent article on the Platonic Number by Dr.
Donaldson (Proc.

of the Philol.

Society).
The conclusions which he draws from these data are summed up by him as follows.

Having assumed that the number of the perfect or divine cycle is the number of the world, and the number of the imperfect cycle the number of the state, he proceeds: 'The period of the world is defined by the perfect number 6, that of the state by the cube of that number or 216, which is the product of the last pair of terms in the Platonic Tetractys (a series of seven terms, 1, 2, 3, 4, 9, 8, 27); and if we take this as the basis of our computation, we shall have two cube numbers (Greek), viz.

8 and 27; and the mean proportionals between these, viz.