36 



SCIENCE 



[N. S. Vol. XXX. No. 758 



required when the legislative and executive 

 authority is vested in experienced educa- 

 tors of the highest order. It might be suflS- 

 cient to say that the strength of the Prus- 

 sian system is due to the fact that for two 

 thirds of a century the work has been cen- 

 tered in a single source of authority for 

 the entire kingdom. 



In American schools considerable loss is 

 due to the fact that during the nine years 

 which correspond to the German Gym- 

 nasium, every pupil must make two 

 changes, viz., from the grades to the high 

 school and thence to college. The work 

 of these nine years is thus carried on under 

 different circumstances, with different 

 names and methods, and under teachers 

 differing in their preparation ; and instruc- 

 tion necessarily suffers in consequence. 



The Germans appreciate the fact that the 

 "teacher can do his best only in an atmos- 

 phere of financial and mental tranquility, 

 rand while insisting on high standards and 

 severe tests at the outset, assure a tranquil 

 career to those who have given evidence of 

 their fitness, by a regular system of pro- 

 motions and pensions. 



In the common schools, the aim is to 

 train good and faithful citizens, the process 

 being called "Erziehung" (bringing up). 



In the higher schools, the aim is to 

 impart learning, and to turn out educated 

 and cultured men. The process is called 

 "Unterricht" (instruction), and leads to 

 privileges and responsibilities before the 

 civil and military law. Attendance upon 

 the common schools is regarded as the duty 

 of those not having better opportunities 

 and is enforced by the state, whereas at- 

 tendance upon a higher school is considered 

 a privilege and the state may restrict the 

 number of persons admitted. 



The work in mathematics done in the 

 higher schools covers practically the same 

 ground as our course to the close of the 



freshman year in college. The general aim 

 of the instruction is facility of calculation 

 with numerical quantities and their appli- 

 cation to the usual circumstances of every- 

 day life, algebra to quadratics, plane and 

 solid geometry, plane trigonometry, the 

 idea of coordinates and some of the funda- 

 mental properties of conic sections. In all 

 of these subjects an effort is made to im- 

 part an intelligent knowledge of the the- 

 orems, as well as skill and facility in their 

 application. 



The entire mathematical education of the 

 boy from the elements of arithmetic to 

 those of analytic geometry takes place in 

 one institution under one management 

 guided by the close supervision of the same 

 director and under the tuition of men of 

 the same scientific training, who are col- 

 leagues, working in close contact, with op- 

 portunities for intimate interchange of 

 ideas. 



The aim of the teaching of arithmetic is 

 to secure facility in operations with num- 

 bers. In order that it may be in harmony 

 with the following algebraic instruction, 

 and prepare for it, the reviews of the fun- 

 damental operations and the treatment of 

 fractions is based upon mathematical form, 

 and the handling of parentheses is con- 

 tinually practised. In fractions the pupil 

 is taught to operate with fractional parts 

 as concrete things, and facility in compu- 

 tation is maintained by continued nu- 

 merical exercises in the algebraic instruc- 

 tion in the following classes.* 



In the Real Gymnasium the scope of in- 

 struction in algebra includes the proof of 

 the binomial theorem, and the solution of 

 equations of the third degree; plane geom- 

 etry, including the theory of harmonic 

 points and pencils and axes of symmetry; 



* For a description of courses see " The Teach- 

 ing of Mathematics in tlie Higher Schools of 

 Prussia," J. W. A. Young, Longmans, 1900. 



