July 3, 1908] 



SCIENCE 



viewed and concluded; solid geometry; 

 practical applications. 



II. Arithmetic and Algebra, 2 hours. 



Arithmetical and geometrical series ; com- 

 pound interest and annuities; quadratic 

 equations ; permutations and combinations ; 

 binomial theorem applied to positive in- 

 tegral exponents. 



UNTEEPRiMA (age, 16-17 years) 



I. Geometry and Trigonometry, 3 hours. 

 Solid geometry continued; theory of 



plane and spherical angles; spherical trig- 

 onometry and its applications to mathe- 

 matical geography; conic sections. 



II. Arithmetic and Algebra, 2 hours. 

 Continued fractions and applications; 



arithmetical series of second order; cubic 

 equations ; problems in maxima and min- 

 ima. 



OBERPRiMA (age, 17-18 years) 



I. Geometry, 3 hours. 



Solid geometry reviewed and concluded; 

 analytic geometry; problems in mathemat- 

 ical geography; geometrical drawing. 



II. Arithmetic and Algebra, 2 hours. 

 Functions and applications to higher 



equations, especially those of the third de- 

 gree; exponential, logarithmic, sine and 

 cosine series; practical applications. 



At the beginning of the nineteenth cen- 

 tury those subjects whose development had 

 been going on through the seventeenth and 

 eighteenth centuries occupied the fore- 

 ground in Germany, namely, Euclidean 

 geometrj'; calculation with letters (Buch- 

 stabenrechnung) ; the theory of logarithms ; 

 the decimal system and the elements of 

 analytic geometry. The elements of differ- 

 ential and integral calculus, although new, 

 were also studied. The general tendency 

 was toward the practical. Mensuration, 

 elementary mechanics and those portions 

 of descriptive geometry which dealt with 



fortifications occupied an important place. 

 It js also noteworthy that a certain amount 

 of mathematical knowledge was considered 

 a preiequisite for philosophical learning, 

 as witness the cases of Leibnitz and Kant. 



Klein divides the nineteenth century into 

 three periods. In the first period, extend- 

 ing from 1800 to 1870, mathematical in- 

 struction was a mixture of the pure and 

 applied. Ideals were high, efforts were 

 directed toward developing individual 

 ability, and attempts were made to teach 

 more than is now required. The candi- 

 date for the position of teacher of mathe- 

 matics must be one who had gone as far as 

 possible into the field, and was himself ca- 

 pable of original research. As the result 

 we find such names as Grassmann, Kum- 

 mer, Pliicker, Weierstrass and Schellbach. 



The second period, extending from 1870 

 to 1890, opened with the victory over 

 France, and the assumption by Germany 

 of a more important international position. 

 This period seemed to be marked by the 

 separation of pure, or abstract, and ap- 

 plied mathematics. In the schools the 

 feeling prevailed that the development of 

 the especially gifted pupil was not so much 

 to be sought as that of the average pupil, 

 and, consequently, greater interest was 

 manifested in methods of instruction. A 

 desire was expressed to replace the early 

 system by a systematic graded course in 

 mathematics, which should keep in view the 

 ability of the constantly developing pupil. 

 Drawings and models were demanded; 

 problems were so stated and aids so given 

 that pupils might see space relations, and 

 not depend so largely upon the logic of the 

 ancient Greeks. This was a direct result 

 of the teachings of Pestalozzi and Herbart. 

 In this period the teaching standard was 

 lowered, as the teacher was only required 

 to possess a knowledge sufficient to work 

 out problems of moderate difficulty. 



