It is evident at once that a great deal of time may be wasted in always following such a plan, perhaps in ever following it consciously. But, on the other hand, probably every good teacher, whether he has heard of Herbart or not, naturally covers these points in substantially this order. For an inexperienced teacher it is helpful to be familiar with them, that he may call to mind the steps, arranged in a psychological sequence, that he would do well to follow. It must always be remembered that there is quite as much danger in "developing" too much as in taking the opposite extreme. A mechanical teacher may develop a new lesson where there is need for only a question or two or a mere suggestion. It should also be recognized that students need to learn to read mathematics for themselves, and that always to take away every difficulty by explanations given in advance is weakening to any one. Therefore, in assigning the new lesson we may say "Take the next two pages," and thus discourage most of the class. On the other hand, we may spend an unnecessary amount of time and overdevelop the work of those same pages, and have the whole lesson lose all its zest. It is here that the genius of the teacher comes forth to find the happy mean. The fourth division of the hour should be reached, in general, in about ten minutes. This includes the recitation proper. But as to the nature of this work no definite instructions should be attempted. To a good teacher they would be unnecessary, to a poor one they would be harmful. Part of the class may go to the board, and as they are working, the rest may be reciting. Those at the board should be limited as to time, for otherwise a premium is placed on mere dawdling. They should be so arranged as to prevent copying, but the teacher's eye is the best preventive of this annoying feature. Those at their seats may be called upon one at a time to demonstrate at the blackboard, the rest being called upon for quick responses, as occasion demands. The European plan of having small blackboards is in many respects better than ours, since pupils cannot so easily waste time. They have to work rapidly and talk rapidly, or else take their seats. What should be put on the board, whether the figure alone, or the figure and the proof, depends upon the proposition. In general, there should be a certain number of figures put on the board for the sake of rapid work and as a basis for the proofs of the day. There should also be a certain amount of written work for the sake of commending or of criticizing adversely the proof used. There are some figures that are so complicated as to warrant being put upon sheets of paper and hung before the class. Thus there is no rule upon the subject, and the teacher must use his judgment according to the circumstances and the propositions.