with a good balance instead of judging them by the hand, we could distinguish the weight of 11 grams from those of 10 and 12 grams, and our formula would become:
A < B, B < C, A < C.
But we should always find between A and B and between B and C new elements D and E, such that
A = D, D = B, A < B; B = E, E = C, B < C,
and the difficulty would only have receded and the nebula would always remain unresolved; the mind alone can resolve it and the mathematical continuum it is which is the nebula resolved into stars.
Yet up to this point we have not introduced the notion of the number of dimensions. What is meant when we say that a mathematical continuum or that a physical continuum has two or three dimensions?
First we must introduce the notion of cut, studying first physical continua. We have seen what characterizes the physical continuum. Each of the elements of this continuum consists of a manifold of impressions; and it may happen either that an element can not be discriminated from another element of the same continuum, if this new element corresponds to a manifold of impressions not sufficiently different, or, on the contrary, that the discrimination is possible; finally it may happen that two elements indistinguishable from a third may, nevertheless, be distinguished one from the other.
That postulated, if A and B are two distinguishable elements of a continuum C , a series of elements may be found, E 1 , E 2 , ..., E n , all belonging to this same continuum C and such that each of them is indistinguishable from the preceding, that E 1 is indistinguishable from A , and E n indistinguishable from B . Therefore we can go from A