340 



SCIENCE 



[N. S. Vol. XXVI. No. 663 



regards series, great difficulty is usually ex- 

 perienced in grasping the idea of con- 

 vergency and divergency at the point where 

 it ordinarily occurs in current text-books, 

 the reason being that it involves the idea 

 of functionality which is of comparatively 

 recent development. Euler first noticed in 

 1748 that convergency of series was neces- 

 sary for computation and partly developed 

 the idea of functionality, but the subject 

 did not receive adequate consideration 

 until demanded by the development of the 

 calculus. 



Two other points may be noted in con- 

 nection with the teaching of algebra. The 

 first is that the graphical method of repre- 

 senting an equation was originated by 

 Descartes, who was also one of the foremost 

 in developing the theory of equations. The 

 inference is that graphs may be advan- 

 tageously used to illustrate the theory of 

 equations, and will also serve as a natural 

 transition to analytic geometry. In this 

 way the historical method meets the ob- 

 jection sometimes raised to our present 

 method of instruction as being conducted 

 in "water-tight compartments." 



The second point is that the examples 

 used to illustrate principles should be so 

 chosen as to stimulate interest, and in order 

 to accomplish this purpose must reflect 

 modern life and local conditions. That 

 this principle of selection was formerly 

 recognized, or at least followed, is shown 

 by some of the time-honored problems 

 which unfortunately still survive. Thus 

 the length of time required to fill or empty 

 a vessel by several pipes had a practical 

 bearing when time was measured by a 

 clepsydra, while such problems as that of 

 the couriers, and the length of time re- 

 quired by several men to complete a piece 

 of AYork, were exceeding useful and inter- 

 esting a century ago, but, now, have no 

 vital interest except perhaps for the his- 



torian. The retention of such problems in 

 modern elementary texts is evidence that 

 the spirit of scholasticism is not yet extinct, 

 and largely accounts for the growing chasm 

 between mathematics and the humanities. 

 Modern life in its growing complexity is 

 teeming with possibilities of mathematical 

 illustration, constantly presenting new 

 problems far greater in cultural value and 

 more wide-reaching in practical significance 

 than any that have yet appeared. To 

 revitalize instruction in elementary mathe- 

 matics the pupil must be taught to recog- 

 nize the true significance of mathematics, 

 as the most powerful instrument yet de- 

 vised by man for ameliorating his physical 

 condition and reconciling cause with effect. 

 Philosophy can never be the proper food 

 for childhood and youth; in elementary 

 instruction the essential feature is that it 

 shall be instinct with life and experience. 



It is beyond the scope of this article to 

 do more than point out the chief features 

 of the historical method and its applica- 

 tion to mathematics. In mathematical 

 pedagogy the present problem is one of ad- 

 justment to modern conditions. This de- 

 mands for its general solution a wide out- 

 look over the history of the past as well as 

 an intimate knowledge of the needs of the 

 present. The routine of teaching too often 

 proves fatal to this breadth of view, leading 

 the teacher into the error of measuring 

 his success by the facility acquired by his 

 pupils in the subject taught. The true 

 criterion of success in instruction ia 

 whether or not it leads the pupil to his 

 highest individual development, refining 

 his spirit and enlarging his field of useful- 

 ness. Like other fine arts, teaching can 

 never be made amenable to fixed rules and 

 rigid methods. There are, however, cer- 

 tain general underlying principles which 

 distinguish the aa-t from pure caprice, and 



