, together with corrective factors whose rôle it would be to ensure invariance. Riemann remarked that in addition to the classical expression, a number of such expressions could be constructed. But if we wish our value of
to be compatible with the existence of the Pythagorean theorem, namely (
for a right triangle), the classical expression of
must be adhered to. For this reason the type of space which is obtained under these conditions is called Pythagorean space; and in what is to follow, we shall have no occasion to consider any other variety.
[28] 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
