est term' from which it can be worked out. The words (Greek)
have been variously translated--'squared and cubed' (Donaldson),
'equalling and equalled in power' (Weber), 'by involution and
evolution,' i.e. by raising the power and extracting the root (as in
the translation). Numbers are called 'like and unlike' (Greek) when the
factors or the sides of the planes and cubes which they represent are
or are not in the same ratio: e.g. 8 and 27 = 2 cubed and 3 cubed; and
conversely. 'Waxing' (Greek) numbers, called also 'increasing' (Greek),
are those which are exceeded by the sum of their divisors: e.g. 12
and 18 are less than 16 and 21. 'Waning' (Greek) numbers, called also
'decreasing' (Greek) are those which succeed the sum of their divisors:
e.g. 8 and 27 exceed 7 and 13. The words translated 'commensurable
and agreeable to one another' (Greek) seem to be different ways of
describing the same relation, with more or less precision. They are
equivalent to 'expressible in terms having the same relation to one
another,' like the series 8, 12, 18, 27, each of which numbers is in the
relation of (1 and 1/2) to the preceding. The 'base,' or 'fundamental
number, which has 1/3 added to it' (1 and 1/3) = 4/3 or a musical
fourth. (Greek) is a 'proportion' of numbers as of musical notes,
applied either to the parts or factors of a single number or to the
relation of one number to another. The first harmony is a 'square'
number (Greek); the second harmony is an 'oblong' number (Greek), i.e. a
number representing a figure of which the opposite sides only are
equal. (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
Tetracty