In mathematics there is probably a deeper cleavage than in any other subject between the real thing, as mathematicians understand it, and the elementary knowledge within the reach of all. "The real thing" is perhaps the most remote and specialised of all branches of learning. For a few it is the best, indeed, the only natural, line of development; but these are few and easily recognised, and even they should not be allowed to specialise too narrowly--that is a point which no one who is not a mathematician will dispute. At the other end of the scale comes the third of the three R's; and about that again there is no controversy, except as to the best methods of teaching it.[2] Yet the schools do not recognise sufficiently clearly this line of cleavage, and many boys who are presumed to have reached the end of the elementary stage remain for some time battering in vain at the doors of the inner temple. These should go back once over the elements again to see if they know them, and then give it up for good. This will mean a cheerful exodus from the upper-middle mathematical divisions. We confess to sympathy with the conservative-radical head master who said, "I shall not advocate the abolition of compulsory Greek in University examinations until I can get people to agree to the abolition of compulsory Algebra." There is perhaps a middle term between elementary and "real" mathematics; that is the mathematics that is the handmaid of physics, and leads us on to the natural sciences. To-day the claims of natural science are very insistent, and they come from more than one quarter. From one quarter comes the claim that science alone of the subjects in the time-table "means business," and makes money, and that in these strenuous times other subjects that lead to mere elegant accomplishments must crowd into a narrow space to make room for the one subject that makes for sheer efficiency. The point is often put with a certain crudity; but we may as well ignore that, and recognise that the just claims of commercial training will have to be met by the schools more fully than heretofore. Only let us recognise commercial training for what it is, and not pretend that it can ever offer a substitute for the liberal education which must continue alongside of it. But the teacher of science will more often take quite other ground, and will claim that his subject, over and above its commercial usefulness, provides most of the ingredients of a Liberal education in itself. He will point to the training it offers in habits of conscientious accuracy, its exemplification of the laws of cause and effect, its undeviating respect for truth, and the inspiration of its endless progress, built up on the heroic researches of the great pioneers.