The few ghost-seers whom we have at present in Germany endeavour naturally to give their mental derangement a scientific colouring, and appeal to individual professors of mathematics and natural science who happen entirely to agree with them, or are supposed to be partially inclined to do so. However, their one sheet-anchor is Zöllner, who is simply a sad proof of the fact that a professorship is no protection from madness; and they can besides, at any rate, point to opportune remarks of Helmholtz and other mathematicians on _n_ dimensions, which they, either intentionally or from mystical weakness of mind, have misunderstood. In an analytical problem the mathematician, instead of one, two, or three dimensions, may place _n_ dimensions without altering thereby the law of the problem and its legitimately resulting corollaries, but it does not occur to him to imagine, under the geometrical expression, ‘nth dimension,’ something given in space, and capable of being apprehended by the senses. When Zöllner gives the well-known example of the inversion of the india-rubber ring which, because only possible in the third dimension, necessarily appeared quite inconceivable and supernatural to a bi-dimensional being, he believes that he facilitates the comprehension of the formation of a knot in a closed ring as an operation practicable in the fourth dimension. In doing this he simply offers one more example of the known tendency of the mystic to delude himself, as he does others, with words which seem to signify something, and which a simpleton is convinced oftener than not that he understands, but which in reality express no idea, and are, therefore, empty sound, void of import.