where the values of the ’s and of and will vary when the mesh-system is changed or when we consider different regions of the same mesh-system. The only case in which the ’s will remain constant is when the mesh-system is of the uniform type, that is, a network of two families of parallel lines intersecting one another. It is a remarkable fact that although everything entering into the expression of varies when we change our mesh-system, yet the value of as defining the value of the square of the distance between two infinitesimally distant points remains unchanged. In other words, is a scalar, an invariant. This allows us to place a new interpretation on the numbers; they appear to act as correctives counterbalancing the variations of and . If we compare the variations in the values of and when the mesh-system is changed, to the advance of a squirrel in a drum, we see that the action of the ’s is similar to a backward revolution of the drum, offsetting the squirrel’s advance, so that the squirrel remains motionless in space.[27] [Pg 91] Now, up to this point we have been considering the expression of a distance between infinitely close points, but in practice we also wish to establish the length of an extended curve traced on the surface. In this case we proceed from point to point along the curve, computing the successive infinitesimal distances , then summating them, or, as it would be more proper to say, integrating them. We thus obtain . Also we may state that the area of an infinitesimal parallelogram formed by two-line elements and is given by Here again we may calculate any finite area by a process of integration, so we see that the finite geometry of the surface can be studied by concentrating our attention on infinitesimal portions and then extending our results from place to place. In short, the method reduces to an application of the differential calculus to geometrical problems, and for that reason is named differential geometry. Powerful as this method of differential geometry has proved to be, there are cases in which it cannot be applied. However, as in the problems of physics with which we shall be concerned, difficulties do not arise, we need not dwell on a number of special cases which in the present state of our knowledge are of interest only to the mathematician.[28]