110 



SCIENCE 



[N. S. Vol. XXVIII. No. 708 



The mathematical requirements for ad- 

 mission are about the same in Germany as 

 with US: algebra, geometry, trigonometry. 

 Not a few students now enter the German 

 engineering college with some knowledge 

 of analytic geometry and even of calculus, 

 but many still come without this knowl- 

 edge. The important point is that the pre- 

 paratory training in mathematics (inclu- 

 ding arithmetic) is distributed systematic- 

 ally and continuously over a period of nine 

 years. The same is true of other prepara- 

 tory studies. It is obviously quite impos- 

 sible to attain in a four-year high-school 

 course the results attained in the nine-year 

 course of a German Gymnasium, Realgym- 

 nasium, or Oberrealschule. This difference 

 in preparation must always be kept in 

 mind in making comparisons between Ger- 

 man and American universities. 



The mathematical courses offered in the 

 German engineering colleges and required 

 for a degree cover plane and solid analytic 

 geometry, differential and integral calculus 

 and differential equations— i. e., about the 

 same subjects that are required in this 

 country. The subject of theoretical me- 

 chanics, which is treated rather differently 

 in different schools, and even in the same 

 school for different degrees, I shall here 

 leave out of consideration, for the sake of 

 simplicity. The amount of time devoted 

 to the higher mathematics, not including 

 mechanics, appears roughly from the fol- 

 lowing table, in which the first figiire in 

 each case gives the number of hours per 

 week devoted to lectures, the second the 

 number of hoiirs devoted to "exercises." 

 These exercises are a comparatively recent 

 innovation. In my time the student had 

 nothing but lectures; to gain a working 

 knowledge of the subject he had to take a 

 text-book and work for himself. Even 

 now, these exercises are optional; they 

 probably exist everywhere, although the 

 table mav not show them. There are no 



periodic examinations such as we have at 

 the end of each semester; but most stu- 

 dents take at the end of their course the 

 Staatsexamen, or if particularly ambitious, 

 the Diplomexamen. The lectures in mathe- 

 matics are rather more advanced and more 

 complete than those in our engineering col- 

 leges. But the requirements in the final 

 examinations are not very high. 



In addition to the more thorough prepa- 

 ration of the German student and to the 

 somewhat higher standard of the lectures 

 on pure mathematics, and largely owing to 

 these circumstances, the treatment of ap- 

 plied mathematics is, I believe, on a higher 

 level in Germany than in this country. 

 The student is better prepared; no time is 

 lost in ' ' recitations, ' ' i. e., in trying to find 

 out whether the student has committed 

 things to memory; the professor is thus 

 enabled to treat scientific questions scien- 

 tifically. Besides, on an average, the Ger- 

 man professor of an engineering subject 

 has himself a higher degree of scientific 

 training and is more interested in the 

 mathematical, and in general the scientific, 

 aspects of his subject than his American 

 colleague. 



It is of course always hazardous and, 

 moreover, of little use to make such gen- 

 eral statements and comparisons; and I 

 do not wish to attach any great importance 

 to them. Neither the German nor the 

 American engineering college is as good as 

 it might be or should be; no institution 

 ever is ; an institution is good only in so 



