The few remarks that I propose to make on this subject will hardly deserve the title of a “paper.” You are doubtless aware of the existence of the International Commission on Mathematical Education. The Commission owes its origin to the distinguished American mathematician, Professor D. E. Smith. At the International Mathematical Congress at Rome three years ago Professor Smith proposed that a “commission” be formed to enquire into questions of teaching, this Commission to report to the next meeting of the Congress, at Cambridge, on August 22-28, 1912. There have been intermediate meetings on the teaching part of the subject, and one meeting was held last September at Milan. It was not a general meeting, being attended mainly by official delegates from different countries. I had the honour of being present there, and it may perhaps interest you if I describe very briefly the matters that were under discussion.

The last half-century has seen a great and significant change in the popular estimation of mathematics. Formerly the subject was regarded as utterly unpractical, and therefore useless in the narrow sense of this term, though it was recognised as providing a training, unique in its character, in logical thought and accurate expression. Now it is regarded, and correctly regarded, as having enormous practical importance in science and engineering. Most, if not all, of those discoveries and inventions which are so profoundly modifying civic and national life have found their origin, or development, or both, in the labours of mathematicians, and this fact is widely known. The mathematician is no longer regarded as a dreamer of dreams; he is classed with the doctor, the engineer, the chemist and all those whose specialised labours have had immense import for the human race.

All who are interested in educational matters cannot fail to be conscious of the widespread spirit of unrest that is abroad. The energy of this association and other similar bodies is proof enough of the increased interest in the efforts that are being made to examine and recast, in the light of modern work, the existing forms of secondary education, and to grapple with the new problems created by a changing system of external forces. Criticism within this sphere is as healthy as it is inevitable, just because it secures a careful scrutiny of the grounds for proposed alterations; and it should serve as a safeguard against fantastic and ill-considered developments of methods and ideas that under proper limitations may be both serviceable and stimulating. Any proposal to introduce a new variety of an old subject, and still more an apparently fresh theme, into the curriculum can only lead to beneficial results if a thorough investigation of the principles involved and the purpose in view has first led to a discrimination between the essential and non-essential, the useful and undesirable, elements of the subject.

Pursuing the comparison with the ordinary methods of Geometry, we may now say that we have established the foundations of Projective Geometry, and have now to make the transition from projective to metrical methods; the distinction between which, though commonly assumed, has never, so far as I am aware, been clearly expressed. The corresponding distinction in the theory of Order is that between unique collation in general, and what I shall now proceed to define as transformation.