When we come to deal with points which do not lie in a straight line, the fundamental region will now be taken to be a plane, or a flat space of three or more dimensions, just as in Chs. I—V it was the straight line. The full discussion will in all cases be given for the plane, in general it will only need small verbal alterations for higher space. Later on the fundamental region may be taken to be a set of points contained in space of a finite number of dimensions, this will include the special case of ordinary curved space. The theory must not be considered to be applicable without fresh investigation to a fundamental space of an infinite number of dimensions; to this question we shall return in the Appendix.

Just as the straight line was to be considered as the geometrical representative of the arithmetic continuum, so the plane is to be regarded as the geometrical representative of the two-fold arithmetic continuum, each point of the plane corresponding uniquely to two numbers in order, (x1, x2), called its coordinates, and conversely each pair of coordinates determining uniquely a point of the plane; the order of the coordinates is formally material, the points (a, b) and (b, a) being different. It will generally be assumed that the coordinates are ordinary rectangular Cartesian coordinates, giving the distances of the point from two perpendicular straight lines, but this is by no means essential, and the idea of coordinates in the plane, or in n-dimensional space, is as independent of the idea of measurement as it was in the straight line.