It is a more difficult question to determine whether the condition here indicated as sufficient to secure the punctual type of convergence is also necessary. The question turns on how far thought-objects of perception possess exact boundaries prior to the elaboration of exact mathematical concepts of space. If they are to be conceived as possessing such exact boundaries, then convergent sets converging to points on such boundaries must be allowed for. The procedure necessary for the specification of the complete punctual condition becomes then very elaborate,[3] and will not be considered here. But such exact determination as is involved in the conception of an exact spatial boundary does not seem to belong to the true thought-object of perception. The ascription of an exact boundary really belongs to the transition stage of thought as it passes from the thought-object of perception to the thought-object of science. The transition from the sense-object immediately presented to the thought-object of perception is historically made in a wavering indeterminate line of thought. The definite stages here marked out simply serve to prove that a logically explicable transition is possible. We accordingly assume that the condition laid down above to secure the punctual convergence of a convergent set of enclosure-objects is not only sufficient, but necessary. It can be proved that, if two convergent sets of enclosure-objects are both equal to a third convergent set, they are equal to each other. Consider now any punctual convergent set (α). We want to define the "point" to which α is a route of approximation in a way which is neutral between α and all the convergent sets which are equal to α. Each of these sets is a route of approximation to the same "point" as α. This definition is secured if we define the point as the class formed by all the enclosure-objects which belong either to α or to any convergent set which is equal to α. Let _P_ be this class of enclosure-objects. Then any convergent set (β) which consists of enclosure-objects entirely selected from members of the class _P_ must be a route of approximation to the same "point" as does the original punctual set α; namely, provided that we choose a small enough enclosure-object in β, we can always find a member of α which encloses it; and provided that we choose a small enough enclosure-object in α, we can always find a member of β which encloses it. Thus _P_ only includes convergent sets of the punctual type, and the route of approximation indicated by any two convergent sets selected from _P_ converges to identical results.