* It is often said that AB and AC subtend equal angles at P : in this form the condition is too narrow, for it excludes the arc BAC of the curve, and too wide, for it admits the parts of the line BC beyond B and C. To include the whole curve some writers say that the angles subtended at P are to be equal or supplementary, but then it is necessary explicitly to remove the whole of the line BC. We may describe the locus by saying that either the angles APB, CPA are equal or the angles APB, APC are supplementary, but surely it is better to learn so emphatic a lesson in the convenience of using crosses. Perhaps the closest accurate approach to the primitive point of view is to say that the crosses (PA, PB) and (PC, PA) are congruent.