Note 27 — Chapter VII · SYSTEMS OF CO-ORDINATES AND DISTANCE

When we consider the ’s in this light, we realise that the expression for might have been anticipated directly without any regard to measurement. We might, for instance, have attempted to construct in a purely mathematical way the possible expressions containing variables such as and , 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.