Nor again can entities of this kind exist separately.For if besides sensible solids there are to be other solids which are separate from them and prior to sensible solids, clearly besides sensible planes there must be other separate planes, and so too with points and lines; for the same argument applies. And if these exist, again besides the planes, lines and points of the mathematical solid, there must be others which are separate;for the incomposite is prior to the composite, and if prior to sensible bodies there are other non-sensible bodies, [20] then by the same argument the planes which exist independently must be prior to those which are present in the immovable solids. Therefore there will be planes and lines distinct from those which coexist with the separately-existent solids; for the latter coexist with the mathematical solids, but the former are prior to the mathematical solids.Again, in these planes there will be lines, and by the same argument there must be other lines prior to these; and prior to the points which are in the prior lines there must be other points, although there will be no other points prior to these.Now the accumulation becomes absurd; because whereas we get only one class of solids besides sensible solids, we get three classes of planes besides sensible planes—those which exist separately from sensible planes, those which exist in the mathematical solids, and those which exist separately from those in the mathematical solids—four classes of lines, and five of points;with which of these, then, will the mathematical sciences deal? Not, surely, with the planes, lines and points in the immovable solid; for knowledge is always concerned with that which is prior. And the same argument applies to numbers; for there will be other units besides each class of points, and besides each class of existing things, first the sensible and then the intelligible; so that there will be an infinite number of kinds of mathematical numbers.

Again, there are the problems which we
enumerated in our discussion of difficulties^{1}: how can they be solved?