THE passage from synthetic projective to metric geometry is seldom made in a systematic and satisfactory manner. Those whose point of view is pure synthetic geometry, for example Reye or B6ger or Enriques, assume the metric geometry as given and confine their attention to interpreting it in terms of projective relations. This fails to bring outthe logical step made in passing from the one to the other. Those who, like Klein and Cayley, approach the subject from the side of non-euclidean geometry introduce the circular points at infinity which even when not directly dependent on analysis are out of the spirit of pure geometry. In any case the start is made from the number-system defined upon a straight line by means of harmonic constructions. Let us start from this point.