The progress of a pupil may be hindered by too much emphasis upon the ground of knowledge. The human mind cannot make an exhaustive study of very many things. Exhaustion is a term applied by logicians to a method of proof in which “all the arguments tending to an opposite conclusion are brought forward, discussed, and proved untenable or absurd, thus leaving the original proposition established by the exclusion of every alternate.” Speaking positively, we may say that exhaustive study of a subject explores it in all its bearings and relations as well as in its nature and essence. In every subject the known is bounded by the unknown; new methods of preparation and investigation constantly reveal novelties in whole classes of objects which it was supposed had been studied exhaustively. The specialist seeks to know all that has been brought to light in his field of research, and to push out the limits of knowledge beyond the goal reached by his predecessors. The thoroughness of the specialist is not required in elementary instruction. The writer knows of a teacher who for an entire term kept a class of boys at work upon highest common factor and least common multiple on the plea that they did not thoroughly understand these subjects. No better plan of disgusting boys with arithmetic and algebra could have been devised. Thorough knowledge of these two subjects involves reasoning and demonstrations more difficult to grasp than half the theorems in Euclid. Instead of aiming at exhaustive treatment, the true teacher is satisfied with knowledge adequate for the subsequent work of the course. If the pupil has reached the stage where he can appreciate the reason why, it may be (though it is not always) wise to raise this question, and to insist on a comprehension of the proof. Very often the mind has enough to do in trying to see _how_; the question _why_ then interferes with the mastery of the mechanical operations. Let any adult take up a system of arithmetic with which he is unfamiliar, say the arithmetic based on counting by fives, or by twelves, or by thirties (each of the last two, mathematically speaking, better than the arithmetic based on tens), he will soon find it is work enough at first for his intellect to perform the operations of adding, subtracting, multiplying, and dividing without reference to the philosophic explanations which exhaustive study would require at every step in the operations.