The discussions which have taken place on the methods of teaching Pure Geometry have been concerned almost entirely with the elementary stages, and the problem of the more advanced teaching has hardly been touched. We should, however, face the phenomenon which occurs time after time—of the boy of undoubted mathematical ability, proceeding to specialise in mathematics, who makes good progress in other branches but who fails to make any real headway in Pure Geometry. There is no doubt that difficulties in the earliest stages, however ably overcome, must of necessity foreshadow further difficulties in the more advanced ones. We propose, therefore, to touch on some points in the senior course. We limit ourselves to the discussion of general points and omit difficulties of detail that arise in the teaching of particular theorems, though we are aware that the problems which arise in the teaching of any section of Pure Geometry cannot be fully solved by examining that section alone. Moreover we do not attempt to discuss the philosophical aspect of the subject, but take the viewpoint of the teacher, whose first desire is to present an effective and interesting weapon for mathematical work.