242 TEE 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 imme- 

 diately 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 differ- 

 ential 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 pre- 

 requisites are usually assigned by the Greek department, but the under- 

 lying 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 pre- 

 requisite 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 Eevolution, 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 require- 

 ments 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, intro- 

 ductory, 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 



