Introduction.—All craniologists are familiar with the form of craniometer or calliper-compasses at present employed for the purpose of ascertaining the various diameters of the cranial box, and they know that these measurements represent certain facts in connection with the breadth, height, and length of the skull as applicable to a number of accepted fixed points upon the surface of the cranium. In effect these diameters are the direct lengths between two points upon an arched surface, i.e., they are the chords of certain arcs, and by the use of the measuring tape we may ascertain the relative proportions between the arc and the chord which subtends it. When several bones contribute to the formation of the arc we may readily determine the relative proportions between each section of the arc ⟨or the chord of each section⟩ and the chord of the entire arc, but it is a much more difficult matter to segment the chord of the entire arc in terms of perpendiculars prolonged to it from definite points upon the arc. For example, by callipers we obtain the glabello-occipital length or chord of the great longitudinal arc, and by measuring tape we may ascertain the frontal, parietal, and supra-inial sections of this arc, but we do not thereby segment the chord of the arc by perpendiculars prolonged to it from the bregma and the lambda. Again, such important transverse diameters of the skull as the minimum frontal, Stephanic, greatest parieto-squamous, and asterionic, are recorded without reference to their proportions on opposite sides of the mesial plane, while the almost constant visible asymmetry of the skull shows that the mesial plane does not necessarily bisect these diameters.