146 



NATURE 



[June 17, 1897 



apply their knowledge instead of increasing the knowledge 

 itself, and altogether to bring the teaching into closer 

 connection with practical needs ; to increase the Konnen 

 as opposed to the Wissen. 



It may be mentioned here that the engineering course 

 in Germany extends over four years, and the students 

 generally enter about the age of eighteen, having, as a 

 rule, spent one year at works beforehand ; so that they 

 do not enter practical life till they have, on an average, 

 reached an age of about twenty-three years or more, as 

 one year of military service has to be taken into account. 

 Even then those who want to enter into the service of the 

 State are tied down by the necessity of passing a final 

 examination at about the age of twenty-seven. 



It is felt that this age is altogether too high, and to 

 remedy this requires, in the opinion of the technical pro- 

 fessors, that the students should be better prepared at 

 school, so that the more elementary part of mathematics 

 and physics can be left out of their course. 



The most active of these technical professors is Prof. 

 Riedler, at Charlottenburg, who is well known in this 

 country as a successful practical engineer. His plans for 

 reform have been published in the Zeitschrift, 1896, and 

 from recent numbers of the same periodical it becomes 

 apparent that his proposals have in the main been carried 

 out at Charlottenburg. In reading through these, one 

 thing is very striking : a great deal of what he writes on 

 the aims of the highest technical education might have 

 been written by Klein in favour of his plans ; in fact, 

 Riedler lays it down as a necessary duty of the technical 

 high schools to care for the highest scientific education 

 of a few engineers. 



He says already in 1895 {Zeitschrift, August 10, 1895), 

 after discussing the needs of the average students, " Die 

 Hochschule muss jedoch mehr bieten, sie muss einer 

 beschrankten Zahl wissenchaftlich Begabter die hochste 

 Stufe Mathematischer Bildung zu erwerben ; aber diese 

 muss eine fruchtbare sein und kann erst dem Fachwissen- 

 schaftlichen studium nachfolgen 



" Die Universitiit Leipzig hat angeblich diese hochst 

 zeitgemasse Aufgabe erfasst, sie ware aber fiir die Tech- 

 nische Hochschule naher liegend. Wo soUen die 

 Universitaten ins gesamt auch nur eine Lehrkraft her- 

 nehmen, die den Ingenieuren das versprochene und Noth- 

 wendige bieten konnte.'" 



Riedler develops his ideas still further in 1896 in several 

 articles {Zeiischrt/t, pp. 301, 337, 374), and also criticises 

 Klein's scheme, of course as a strong opponent. 



From these quotations it is clear that he feels the same 

 need for the highest technical education as Klein ; but 

 we may well turn his question round and ask where will 

 he find a professor who can teach the highest mathematics 

 to technical students, and who is at the same time a 

 practical engineer? Here Klein comes in ; his plan will 

 help to educate such men. 



Riedler has in these papers a fling at higher mathe- 

 matics as cultivated at the Universities, where ".Abel and 

 Riemann " count much, where one lives in regions in which 

 ^-functions disappear, and in hypo- or meta-geometry, 

 where " dimensions cease and manifoldnesses begin," and 

 where the student learns "gymnastics in four dimensions." 



It is easy from a "practical" point of view to make 

 light of many parts of pure mathematics. Even Klein 

 NO. 1442, VOL. 56] 



professes that he has often had doubts whether those 

 theories in which the mathematician delights are really 

 worth the tro'jhV of increasing and developing, and in 

 his opinion they would not be if they should never be of 

 use in application to physical and engineering problems. 



But, he adds, he has always come away from such con- 

 templations with the conviction that his (and all mathe- 

 maticians') optimistic views are justified by the belie 

 that they will assist in due time in subjugating nature to 

 science. In fact, the history of science is so full of 

 examples that it is unnecessary to quote any. 



If Riedler should succeed in getting the influence of 

 all those speculations banished from the Hochschule 

 these will soon become as fossilised as the Universities, in 

 his opinion, are at present. 



But let us return to Klein's plans. His chief idea, as 

 gathered from the various papers and from his own ex- 

 pressions during his recent visit to England, can be 

 stated briefly enough. 



He wants to bring together again theory and practice ; 

 he wishes the Universities to take their proper place in 

 modern German life, which differs from that of former 

 days by its enormous energy in industry and commerce. 

 He acknowledges that much has been done outside the 

 Universities, and for purely practical purposes, to develop 

 science ; that altogether new methods of investigation 

 have been invented by engineers, and not only in physical 

 investigation but even in mathematics. 



He wishes to introduce these at the Universities to 

 enable them to fulfil their duties properly, and he hopes 

 by thus raising these institutions to enable them so to 

 develop science that the results will be practically useful 

 and repay the debt to engineers which science now owes 

 them. 



But he also hopes greatly to improve the education of 

 matliematical and science teachers. There can be no 

 doubt that the chief education of these must and will 

 remain in the hands of the Universities — that they cannot 

 be left to the technical schools. It is quite impossible to 

 estaljlish schools exclusively for future engineers, because 

 it is impossible to settle at an early age what career a 

 boy will select, and the majority of schools must be of a 

 general character. 



But what is highly desirable is that all teachers, not 

 only the scientific ones, should be better prepared in 

 science, and should gain a higher idea of the value of 

 practical work, and that the old spirit according to which 

 education can only be gained through the old languages, 

 should be broken. 



Not only the would-be engineers, but all boys must be 

 made to feel this change, and therefore all teachers must 

 be imbued with modern notions. To have some educated 

 at the University and some at the technical schools would 

 intensify the existing antagonism between science and 

 classics. This ideal education of teachers can only be 

 obtained at the Universities, and here only if these are 

 themselves modernised. Klein's scheme tends to bring 

 this about ; it does not exclude the possibility of some 

 teachers of science and mathematics being educated 

 at high schools, nor that many of these should spend 

 a part of their time in studying at a high school; it 

 would be easy to devise a plan by which this could be 

 accomplished. 



