is the dimensionality of the continuum.
It is a remarkable fact that when, and only when, the continuum is flat, this expression for
can break up into a sum or difference of squares. When it breaks up into a sum of squares, the continuum is called Euclidean; when into sums and differences of squares, it is called semi-Euclidean, and is said to have positive and imaginary dimensions; but in all such cases it remains flat. (Space-time is an illustration of this latter species of continuum.)
And now suppose we wish to discover the equations of geodesics, that is to say, of the straightest of lines compatible with the structure of the continuum. We obtain a geodesic between two points
and
by expressing the fact that the line stretching between