I have contended that, on the contrary, nothing would force the Spherelanders to assume a third dimension, while they would find it impossible exactly as we find a fourth impossible--not logically, that is to say, but only as a presentable construction in given space. After a somewhat elephantine piece of humour, about socialistic whales in a four-dimensional sea of Fourrier's _eau sucrée_, Lotze proceeds to a proof, by logic, that every form of intuition, which embraces the whole system of ordered relations of a coexisting manifold, _must_ have three dimensions. One might object, on _à priori_ grounds, to any such attempt: what belongs to pure intuition could hardly, one would have thought, be determined by _à priori_ reasoning[107]. I will not, however, develop this argument here, but endeavour to point out, as far as its obscurity will allow, the particular fallacy of the proof in question. [Illustration] Lotze's argument is as follows. In this discussion, though our terminology is necessarily taken from space, we are really concerned with a much more general conception. We assume, in order to preserve the homogeneity of dimensions, that the difference (distance) between any two elements (points) of our manifold--to borrow Riemann's word--is of the same kind as, and commensurable with, the difference between any other two elements. Let us take a series of elements at successive distances _x_ such that the distance between any two is the sum of the distances between intermediate elements. Such a series corresponds to a straight line, which is taken as the _x_-axis. Then a series _OY_ is called perpendicular to the _x_-axis _OX_, when the distances of any element _y_, on _OY_, from +_mx_ and -_mx_ are equal. By our hypothesis, these distances are comparable with, and qualitatively similar to, _x_ and _y_. So long as _OY_ is defined only by relation to _OX_, it is conceptually unique. But now let us suppose the same relation as that between _OX_ and _OY_, to be possible between _OY_ and a new series _OZ_; we then get a third series _OZ_ perpendicular to _OY_, and again conceptually unique, so long as it is defined by relation to _OY_ alone. We might proceed, in the same way, to a fourth line _OU_ perpendicular to _OZ_. But it is necessary, for our purposes, that _OZ_ should be perpendicular to _OX_ as well as _OY_. Without this condition, _OZ_ might extend into another world, and have no corresponding relation to _OX_--this is a possibility only excluded by our unavoidable spatial images. At this point comes the crux of the argument. _That OZ_, says Lotze, which, besides being perpendicular to _OY_, is also perpendicular to _OX_, must be among the series of _OY_'s, for these were defined only by perpendicularity to _OX_. _Hence_, he concludes, there can only be even a third dimension if _OZ_ coincides with one, and--as soon as _OX_ is considered fixed--with _only_ one, of the many members of the _OY_ series.