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. A similar verification would ensue if we were to effect measurements along a circumference and its diameter. Thus, if at a central point on the earth’s surface we fix one end of a rope and cause the other extremity to rotate around the central point, it will describe a circle on the earth’s surface. We shall then find, upon measuring the length of the circumference marked out by the extremity of the rope, that the ratio of the length of this circumference to the length of the diameter is a variable number, always smaller than and depending on the area covered by our circle; whereas in Euclidean geometry this ratio would be the constant number . There is still another result of importance which must be mentioned. On a plane, the farther we wander along a straight line, the farther we move from our starting point, whereas on our sphere we see that if we follow a straight line or geodesic we shall finally return to our starting point after having circled the earth. We express this fact by saying that in Riemann’s geometry space is finite, yet unbounded. It is finite since we cannot wander away indefinitely from our starting point; it is unbounded because however far we go, we never come to a stone wall or a gap beyond which we can proceed no farther. On the other hand, Euclidean space, such as that of the plane surface, is infinite and unbounded.