neither reference to the fourth dimension, nor any kind of internal contradiction. This fact disproves Lotze's contention, which arises solely from inability to divest his imagination of Euclidean ideas. Lotze next attacks Helmholtz for the assertion that _B_ would know nothing of parallel lines--parallel _straight_ lines, as the context shows, he meant to say[104]. Lotze, however, takes him as meaning, apparently, mere curves of constant distance from a given straight line, which are part of the regular stock-in-trade of Metageometry. Parallels of latitude, in the geographical sense, would not--with the exception of the equator--appear to _B_ as straight lines, but as circles. _Great_ circles he _would_ call straight, and this fact seems to have misled Lotze into thinking _all_ circles were to be treated as straight lines. Parallels of latitude, therefore, though _B_ might call them parallels, would not invalidate Helmholtz's contention, which applies only to straight lines. The argument that such small circles would be parallel, which we have just disposed of, is only the preface to another proof that _B_ would need a third dimension. Let us call two of these parallels of latitude _l{n}_ and _l{s}_, and let them be equidistant from the equator, one in the northern, one in the southern hemisphere. Consecutive tangent planes, along these parallels, converge, in the one case northwards, in the other southwards. Either _B_ could become aware of their difference, says Lotze, or he could not. In the former case, which he regards as the more probable, he easily proves that _B_ would infer a third dimension. But this alternative is, I think, wholly inadmissible. Tangent planes, like Euclidean planes in general, would have no meaning to _B_; unless, indeed, he were a metageometrician, which, with all his metaphysical and mathematical subtlety, the argument supposes him not to be--and to such a supposition Lotze, surely, is the last person who has a right to object. Lotze's attempted proof that this is the right alternative rests, if I understand him aright, on a sheer error in ordinary spherical Geometry. _B_ would observe, he says, that the meridians made smaller angles with his path towards the nearer than towards the further pole--as a matter of fact, they would be simply perpendicular to his path in both directions. What Lotze means is, perhaps, that all the meridians would meet sooner in one direction than in the other, and this, of course, is true. But the poles, in which the meridians meet, would appear to _B_ as the centres of the respective parallels, while the parallels themselves would appear to be circles. Now I am at a loss to see what difficulty would arise, to _B_, in supposing two different circles to have different centres[105]. We must, therefore, take the first alternative, that _B_ would have no sort of knowledge as to the direction in which the tangent planes converged. Here Lotze attempts, if I have not misunderstood him, to