Much has been said about the danger of dividing the various branches of pure mathematics into “water-tight compartments,” and books have been written with a view to breaking down from the very beginning the barriers between the different subjects.

I feel, however, that to the average boy who has not yet reached Matriculation standard, the division of Mathematics into Arithmetic, Algebra and Geometry is very real and most helpful. A straight-forward question inmensuration, the study of the identity (a+b)2=a 2+2ab+b 2, and the theorem that the square on a line is equal to the sum of the squares on its two segments together with twice the rectangle contained by them, for example, will and should appear as separate questions—the first being linked with other similar exercises in Arithmetic, the second depending on algebraical multiplication and being a study in form rather than in area, and the third turning on a geometrical construction. Only after each has been treated in the appropriate way should we show the pupil the cross-connections existing in the three problems.