There is still another point which we must mention. It might be argued that in Riemann’s geometry of two dimensions, that is to say, in the geometry of the spherical surface when Euclidean measurements are adhered to, a straight line on the sphere, being a great circle, constitutes a closed curve, whereas in Euclidean geometry a straight line can never constitute a closed curve. Here, however, the difficulty which arises does not pertain so much to Euclideanism and non-Euclideanism as to another branch of geometry, Analysis Situs, on which we shall make a few brief remarks in a note at the end of the chapter. The fact is that the geometry of the cylinder embedded in Euclidean space is Euclidean to the same extent as the geometry of the plane, and yet on a cylinder certain straight lines, namely, the rings which surround the cylinder, constitute closed curves. It might be objected that these rings are not Euclidean straight lines. But here the critic would be confusing geometry and dimensionality. From the standpoint of two-dimensional Euclidean geometry, the rings on the cylinder are perfectly straight lines, and it would never enter the mind of the flat being moving over the cylindrical surface to view them in any other light. Only when we represent the cylinder’s surface in a three-dimensional Euclidean space must we regard the rings as curves. In short, lines that would be straight in two dimensions might be curved when viewed from the standpoint of a higher dimensionality. By analogy, lines which would be regarded as straight in three-dimensional space might be recognised as curved were we to appeal to a fourth [Pg 64] dimension. Thus, whichever way we choose to investigate the concept of straightness, we find that it eludes us more and more. It appears unnecessary to dwell further on these rather special problems, for they would lead us too far afield. Now, finally, there is still another reason, on which we have lightly touched, which must make us very wary of being led astray by faulty analogies. When, for instance, we consider the two-dimensional geometry of a spherical surface embedded in a three-dimensional Euclidean space, we realise that the surface divides space into an outside and an inside. We should then be inclined to argue that a three-dimensional Riemannian or spherical space must also divide four-dimensional space into an inside and an outside.