I doubt, however, whether it is desirable to plunge the student into the full rigour of euclidean geometry without some mitigation. It is for this reason that the modern habit, at least in England, of laying great stress in the initial stages on training the pupil in simple constructions from numerical data is to be praised. It means that after a slight amount of reasoning on the euclidean basis of accuracy, the mind of the learner is relieved by doing the things in various special cases, and noting by rough measurements that the desired results are actually attained. It is important, however, that the measurements be not mistaken for the proofs. Their object is to make the beginner apprehend what the abstract ideas really mean. Again in algebra, the notation and the practical use of the symbols should be acquired in the simplest cases, and the more theoretical treatment of the symbolism reserved to a suitable later stage. My rule would be initially to learn the meaning of the ideas by a crude practice in simple ways, and to refine the logical procedure in preparation for an advance to greater generality. In fact the thesis of my paper can be put in another way thus, the object of a mathematical education is, to acquire the powers of analysis, of generalisation, and of reasoning. The two processes of analysis and generalisation were in my previous statement put together as the power of grasping abstract ideas. But in order to analyse and to generalise, we must commence with the crude material of ideas which are to be analysed and generalised. Accordingly it is a positive error in education to start with the ultimate products of this process, namely the ideas in their refined analysed and generalised forms. We are thereby skipping an important part of the training, which is to take the ideas as they actually exist in the child's mind, and to exercise the child in the difficult art of civilising them and clothing them. The schoolmaster is in fact a missionary, the savages are the ideas in the child's mind; and the missionary shirks his main task if he refuse to risk his body among the cannibals. At this point I should like to turn your attention to those pupils forming the mathematical section. There is an idea, widely prevalent, that it is possible to teach mathematics of a relatively advanced type--such as differential calculus, for instance--in a way useful to physicists and engineers without any attention to its logic or its theory.