 PM concludes with a seemingly dangling excursion into the special case of the measurement of “Cyclic Families”. For “such cases as the angles at a point, or the elliptic straight line, we require a theory of measurement applicable to families which are not open” (PM, Volume III, 457). The angles at a point will be measured from 0 to 360 degrees, and then start over again at 0 to measure an object rotating around a point. The many ratios that represent the rotations are represented by a “principal ratio”, which is used to assign the measurement of degrees.