As I can pretend to only limited practical experience in the matter of teaching the foundations of Mathematics, and of Geometry in particular, I have thought that the time allowed for my remarks may be advantageously occupied in a rapid survey of what the geometrical method has accomplished, and the estimation in which it has been held at different periods of the history of the development of Mathematical Science. Thus, avoiding expert questions relating to Education, we may begin by considering from a general standpoint what is implied in the science of Geometry, for the right ordering of which, in its educational relations, this Association has been working for a long series of years. It would, I presume, be against the instincts of our members to interpret the subject in any narrow sense. Geometry is usually based on the partly mathematical, partly logical, framework which has descended to us from Greek times, and which is chiefly associated with its earliest great systematization in the work of Euclid. But modern development would perhaps render any definition of its scope incomplete that did not include all methods in formal and physical Mathematics which proceed by direct contemplation of the relations of the subjects considered, instead of through representations of them that are merely quantitative or numerical.