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'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.