JuxT 3, 1908] 



SCIENCE 



The ground to be covered depends 

 largely upon the ideals of the school. Al- 

 though the formal side must not be 

 neglected and a thorough knowledge of 

 processes must be obtained, the principal 

 aim is ta give a clear conception of the 

 fundamental ideas and their meaning. 



Much confusion often results from the 

 fact that a word possesses several mean- 

 ings. Thus a purist might define ele- 

 mentary mathematics as those parts of the 

 subject in which the conception of a limit 

 is avoided. The more commonly accepted 

 definition of elementary mathematics, 

 however, admits the idea of limits but ex- 

 cludes the special forms represented by the 

 symbols dy/dx and fydx. Neither defini- 

 tion can be made to agree with the practise 

 of the schools. For example, the first 

 definition would exclude the consideration 

 of such irrationals as V 2, and tt used in 

 determining the area of a circle as the 

 limit approached by a polygon. On the 

 other hand, the second definition might be 

 made to include much which does not be- 

 long in the schools, as, for example, the 

 so-called "elementary" theory of analytic 

 functions of the complex variable. The 

 first definition might also be made to in- 

 clude much of the most difficult nature, 

 such as advanced portions of the .theory 

 of numbers. In geometry there is also a 

 new use of the word elementary. That 

 portion of geometry is now styled ele- 

 mentary which is based on the Euclidean 

 or ancient Greek geometry, the simplest 

 conceptions of the newer geometry being of 

 too severe a nature for the schools. 



The only definition which will hold 

 within the schools is a very practical one, 

 namely, that shall be called elementary in 

 the various branches of mathematics which 

 can be grasped by the average pupil with- 

 out extraordinary effort of long duration. 



The material which constitutes ele- 



mentary mathematics varies with time; 

 that is to say, it is subject to the law of 

 historical delay. Subjects which formerly 

 were not considered elementary have, by 

 unproved processes of instruction, been 

 made so, as is shown, for instance, in the 

 geometry of the ancients. If, in conse- 

 quence of the above definition, the extent 

 of the field of elementary mathematics be- 

 comes too great and indeterminate, it comes 

 within the province of the schools to choose 

 those parts which best serve their purpose. 



Mathematical instruction, on the level at 

 which it is at present carried on in the 

 upper classes of the higher schools, has 

 existed in Germany since about the begin- 

 ning of the eighteenth century. Chris- 

 tian Wolf, who was professor at Halle and 

 one of the foremost schoolmen of this 

 period, included in his list of elementary 

 mathematics, in addition to the geometry 

 of the ancients, a great many of what were 

 at that time modern achievements, such as 

 calculations with letters, negative numbers, 

 algebraic equations and logarithms ; in fact, 

 practically everything which was known to 

 mathematicians in 1700. It is evident that 

 calculus was not included, for at that time 

 the knowledge of calculus was the posses- 

 sion of only a few investigators of the 

 highest type, whose efforts were not so 

 much directed toward the clearing up of 

 fundamental principles as toward the solu- 

 tion of new and difficult problems. To the 

 layman, calculus seemed a sort of witch- 

 craft. Cauchy 's great work on differential 

 and integral calculus appeared in 1821, 

 but the schools had already been led into 

 certain channels, and it was not possible 

 to divert then toward a subject which was 

 only in process of formation. 



Moreover, it is true in general that 

 mathematics is more susceptible than any 

 other subject to hysteresis. A new idea 

 finds its way into the schools through the 

 lectures of university professors. A new 



