* If we were not illustrating the generalization of the ordinary metrical proof, we could say simply : (1) As AFBR and AECQ are harmonic ranges, the points P, E, F are in line; (2) Thence, by definition, AO contains the harmonic conjugate, D, of P, in regard to B and C. And this remark shews up nicely the cumbrousness of the metrical theory of mid points. For the two results; (1) that the join of the mid points of two sides of a triangle is parallel to the third side; (2) that if in a quadrilateral (trapezium) BCEF, the opposite sides BC, FE be parallel, while BF, CE meet in A, and BE, CF meet in 0, then AO bisects BC undFE, do not lie at the beginning of the subject. To a boy who has appreciated the point of view here propounded these results would be obvious.