SCIENCE 



[N. S. Vol. XXVIII. No. 705 



generation of teachers is thus trained who 

 give the idea shape in their class work, 

 until finally it becomes the common 

 possession of all. In accordance with this 

 process of development, Klein expresses 

 his belief that it is now time to make the 

 fundamentals of calculus a necessary part 

 of elementary instruction. To illustrate 

 the historical development of the subject, 

 he quotes the words of his teacher of 

 mathematics, who said in the fall of 1865, 

 "In elementary mathematics we can prove 

 things, but in the higher mathematics it is 

 different. They resemble a philosophical 

 system, which we may or may not believe. ' ' 

 It is remarkable that this idea has com- 

 pletely disappeared in such a short in- 

 terval. 



For a long time calculus was regarded 

 with distrust, but as it received recogni- 

 tion in the official course of study of 1900, 

 Klein believes that it is time to take ad- 

 vantage of this favorable attitude to put 

 that which has taken centuries for prepara- 

 tion upon a general and recognized basis. 

 As a matter of fact the fundamental ideas 

 underlying the calculus are actually taught 

 in many schools. In a few Ober-Real 

 schools they are regularly taught as cal- 

 culus, but in the majority of the schools 

 they are given in a very roundabout man- 

 ner. It amounts to this, that students are 

 actually taught to differentiate and inte- 

 grate as soon as occasion for the same 

 arises, but the terms differential and in- 

 tegral are avoided. 



An inspection of the text-books in cur- 

 rent use in the higher schools shows con- 

 clusively that many of the simpler ideas 

 of calculus are in use, but are rendered 

 more or less difficult of comprehension by 

 the avoidance of symbols and operations, 

 which, if understood, would render the 

 work comparatively easy. If the field of 

 physics were examined, instances of this 

 kind would be greatly multiplied, especially 



in the fundamentals of mechanics and 

 electrodynamics. Evidently, then, calcu- 

 lus occupies a more extensive field than is 

 commonly supposed, but it is taught un- 

 systematically, and is merely tacked on 

 here and there to the general content of 

 mathematical instruction. Klein is of the 

 opinion that instead of making instruction 

 in calculus in those grades whose work 

 demands its employment merely incidental, 

 desultory and generally unsatisfactory, it 

 should be made the central idea of all 

 instruction, and the other ideas and work 

 grouped around it. 



At present calculus is made the begin- 

 ning of higher mathematics and is accom- 

 panied by a revolution in thinking. This 

 revolution furnishes good evidence of the 

 aimlessness of the earlier instruction as 

 contrasted with the ideas with which the 

 pupil later comes in contact. Klein's sug- 

 gestion aims to spare the pupil this sud- 

 den change, by gradually accustoming him 

 to the methods of thinking which prevail 

 in his later work. 



The traditional methods of teaching will 

 readily accommodate themselves to this new 

 idea, and in fact will be much simplified 

 thereby. This statement is borne out by a 

 comparison of the cumbrous algebraic 

 method of solving problems with the 

 methods of calculus. On the other hand, 

 no harm is done if certain portions of 

 mathematics which supposedly have merely 

 a formal training value, such as artificial 

 equations solved by quadratic roots, and 

 trigonometric analysis, are pushed to the 

 rear, for the new material gives ample op- 

 portunity for formal work. 



The inadequacy of the present system is 

 clearly shown in the education of the 

 lawyer, physician or chemist. As regards 

 the first two, it is, of course, understood at 

 the outset that their work in mathematics 

 must necessarily be brief, as their major 

 subject allows little time for it. Hence 



