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 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.
This line of reasoning would be correct in the present case, but it is important to note that this conception of an inside and an outside is in no wise essential to non-Euclidean geometry. Had we adopted the more rational method of presenting three-dimensional non-Euclidean space as due to the peculiar behaviour of our measuring rods, the conception of an inside and an outside would have been meaningless. We must understand that this notion arose merely as a result of our particular choice of a mode of presentation, and in no wise constitutes an intrinsically necessary condition.
If these rather delicate points have been understood, no harm can be done by discussing non-Euclidean geometries as the geometries obtained by applying Euclidean rods on curved surfaces. In many respects, indeed, this method of presentation is helpful; for accustomed as we are in everyday life to effect measurements with Euclidean rods, we are able to visualise more easily a series of abstract investigations when this familiar procedure is followed.
Let us now proceed to a more thorough study of the preceding method. We have said that non-Euclidean results are obtained when we confine ourselves to Euclidean rods while conducting our measurements on curved surfaces. In the case of Riemannian geometry of two dimensions, the required surface is that of a sphere embedded in three-dimensional Euclidean space. In the case of Lobatchewski’s geometry, it is a saddle-shaped surface known as a pseudosphere (subject to the limitations mentioned previously).
We will confine ourselves to the study of Riemann’s geometry for the time being. Such a study in two dimensions permits easy visualisation, owing to the fact that since the earth is spherical in shape it is precisely this type of geometry which we obtain when we conduct measurements on its surface with rigid Euclidean measuring rods (assuming the surface to be perfectly smooth, like that of the ocean on a calm day). In fact, we shall see that the postulates of Riemann’s geometry are immediately verified on the sphere.
To begin with, the shortest distance, or, better still, the most direct distance, between two points on the earth’s surface, when Euclidean rigid rods or inextensible tape measures are employed as a means of measurement, is an arc of a great circle. Great circles, such as meridians on the surface of our planet, constitute therefore the straight lines or geodesics of Riemann’s geometry; and when discussing Riemann’s geometry we may call them straight lines.
Consider, then, one such straight line or geodesic, and let it be that particular great circle which we call the equator. According to the axioms of Euclid’s geometry two perpendiculars drawn at two different points of the same straight line can never meet; they lie parallel to one another. On the sphere, on the other hand, two perpendiculars to the equator are two meridians and these meridians always meet at the pole. In fact it is impossible to draw any two great circles on the sphere which do not intersect. Hence we see that no parallel geodesics can exist in Riemann’s geometry. Likewise, whereas in Euclid’s geometry the sum of the three angles of a triangle is equal to two right angles, in Riemann’s geometry this sum is always greater than two right angles. This we could easily verify by stretching ropes between three distant points on the earth’s surface so as to form a triangle, and then summing the values of the angles at the three corners of our triangle.