For the geometry of a space of more than two dimensions to be determined fully, we must state the values of the components of the tensor (mentioned in the note); there being twenty such components at every point for a four-dimensional space. We may note that, contrary to the Gaussian curvature , is not invariant. It splits up into components each one of which may vary in value when the mesh-system is changed. Yet, in spite of this variability, still defines the curvatures of the space, hence still refers to something that is intrinsic and irrelevant to our choice of mesh-system. Such is one of the characteristics of a tensor. This particular tensor of twenty components (in a four-dimensional space) is known as the Riemann-Christoffel tensor.[33] Curiously enough, it was discovered by Riemann, not when investigating the geometry of space, but when considering a problem in heat. Only when every one of these twenty components of the tensor vanishes at every point is it possible to assert that the four-dimensional space is Euclidean, or at least flat. Thus, whereas for two dimensions Euclideanism was ensured when one condition was satisfied at every point, namely, the vanishing of the Gaussian curvature, , on the other hand, when we step up to four dimensions, twenty conditions are needed, and these are given by the vanishing at every point of the twenty expressions which in their aggregate constitute the Riemann-Christoffel tensor . We may also mention that in addition to the Riemann-Christoffel tensor , Einstein has made use of the other tensor (previously discovered by Ricci), which in four-dimensional space is defined by ten separate relations between the values of the [Pg 98] ’s at a point and their values at neighbouring points. The vanishing of this other tensor at every point, that is to say, the vanishing of the ten new component expressions at every point, is insufficient in itself to ensure the Euclideanism of the space; although, of course, this vanishing imposes certain restrictions on the space’s non-Euclideanism. These restrictions are less severe than those defined by the vanishing of the twenty components of the Riemann-Christoffel tensor, which ensures perfect flatness; on the other hand, they are more stringent than those imposed by a mere vanishing of the Gaussian curvature .[34]