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.