Page:ScienceAndHypothesis1905.djvu/63

 senses—or, if the reader prefers, from the rough results of Fechner's experiments; I have shown that these results are summed up in the contradictory formulae:—A = B, B = C, A < C.

Let us now see how this notion is generalised, and how from it may be derived the concept of continuums of several dimensions. Consider any two aggregates of sensations. We can either distinguish between them, or we cannot; just as in Fechner's experiments the weight of 10 grammes could be distinguished from the weight of 12 grammes, but not from the weight of 11 grammes. This is all that is required to construct the continuum of several dimensions.

Let us call one of these aggregates of sensations an element. It will be in a measure analogous to the point of the mathematicians, but will not be, however, the same thing. We cannot say that our element has no size, for we cannot distinguish it from its immediate neighbours, and it is thus surrounded by a kind of fog. If the astronomical comparison may be allowed, our "elements" would be like nebulae, whereas the mathematical points would be like stars.

If this be granted, a system of elements will form a continuum, if we can pass from any one of them to any other by a series of consecutive elements such that each cannot be distinguished from its predecessor. This linear series is to the line of the mathematician what the isolated element was to the point.