Suppose that a sequence of discs
arranged in decreasing order of diameters, forms a packing within the unit plane square I2. It has been shown, by Florian(1), that the area of
is at least O(a), where a is the radius of θn. However, Gilbert (2) has produced some empirical results for the Apollonius packing 71 of discs which seem to suggest that for such a packing, the area of the set
is at least O(as) for some positive real number s, less than one. As Gilbert remarks, it is difficult to imagine that the Apollonius packing is not the extremal case, and so, that it would seem likely that there exists a positive real number s, less than one, such that for a general packing, the area of
is at least O(as). The purpose of this paper is to establish this result by showing that 0·97 is an allowable value for s.