*THE POPULAR SCIENCE MONTHLY*

matics enters more or less into every engineering curriculum. Consider as an example that of the electrical engineer. Differential equations are frequently met in the study of electricity, and hence a mathematical course in this subject is a prerequisite. The sequence of subjects immediately prerequisite to this study, given in descending order, is integral calculus, differential calculus, analytical geometry, college algebra, trigonometry, solid geometry, plane geometry, elementary algebra and arithmetic. These are actual prerequisite courses and a study of differential equations requires a knowledge and ready facility in all of them. As commonly taught, and except for plane and solid geometry, there are few daily lesson assignments the subject matter of which does not enter directly into the later study of differential equations. The penalty for slip-shod work in the early courses is sure and cumulative. Consider, upon the other hand, the study 'in college of the books of Herodotus, the plays of Sophocles, or the orations of Demosthenes. Certain prerequisites are usually assigned by the Greek department, but the underlying idea is generally to prescribe sufficient elementary Greek to assure a homogeneous class with a fair facility in the language. Slipshod work in earlier courses does not bear so immediate and evident a penalty as in the case of differential equations. The entire omission of some prerequisite as Homer or certain dialogues of Plato would not seriously inconvenience a student. In fact, no particular day's lesson in most of the earlier courses may be said to be absolutely prerequisite.

Many illustrations similar to the one just cited are to be found in engineering curricula. A further illustration in arts may well be given. Thus consider history. A general introductory course in medieval and modern history is usually a prerequisite to further study. But later courses dealing with special periods as that of the French Revolution, do not demand so imperatively exact and complete preparation in the introductory courses.

The technical school, with its groups of continuous and interdependent courses, offers more severe mental discipline for the average student than does the arts college, where the prerequisites are largely formal requirements for the sake of a logical continuity that lays small burden on the student's mental powers. This is true even under a group elective system. Thus imagine a group requirement which called for four courses in the same scientific department. Biology offers general, introductory, and but slightly related courses in physiology, anatomy, botany and zoology. The disciplinary value of a general introductory course in any subject, which is not followed by a punishing and detailed study to which it is immediately prerequisite, is slightly more than that of a popular series of illustrated lectures with collateral reading.

The aphorism of pedagogy, "No impression without expression," may be extended to read "There is no enduring impression of an unused