Note 28 — Chapter VII · SYSTEMS OF CO-ORDINATES AND DISTANCE

In a very brief way the difficulties are as follows: Differential geometry involves continuity; hence in a discrete continuum it would lose its force. But even this is not all, for there are various kinds of continuity, and continuity must be of a special type for differential geometry to remain applicable. For instance, in the foregoing exposition of the method we assumed that the expression of would tend to a definite limit when the points were taken closer and closer together. In particular we assumed that for infinitesimal areas of the surface the Pythagorean theorem would hold. This was equivalent to stating that infinitesimal areas of our curved surface could be regarded as flat or Euclidean, hence as identified with the plane lying tangent to them. Inasmuch as the restriction of Euclideanism in the infinitesimal is precisely one that Riemann has imposed on space, we need have no fear of applying the differential method to the types of non-Euclidean space thus far discussed.

Nevertheless, we may conceive of spaces where would not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within waves ad infinitum. The situation would be similar to that presented by curves with no tangents at any point. However, such cases are only of theoretical interest.