412 



SCIENCE. 



[N. S. Vol. XVII. No. 428. 



practical as well as theoretic, of the funda- 

 mental methods of science. 



In connection with what has already been 

 said, the general suggestions I now add 

 will, I hope, be found of use when one 

 enters upon the questions of detail involved 

 in the organization of the course. 



As the world of phenomena receives at- 

 tention by the individual, the phenomena 

 are described both graphically and in terms 

 of number and measure; the number and 

 measure relations of the phenomena enter 

 fundamentally into the graphical depiction, 

 and furthermore the graphical depiction of 

 the phenomena serves powerfully to illu- 

 minate the relations of number and meas- • 

 ure. This is the fundamental scientific 

 point of view. Here under, the term 

 graphical depiction I include representa- 

 tion by models. 



To provide for the needs of laboratory 

 insti'uction, there should be regularly as- 

 signed to the subject two periods, counting 

 as one period in the curriculum. 



As to the possibility of effecting this 

 unification of mathematics and physics in 

 the secondary schools, objection will be 

 made by some teachers that it is impossible 

 to do well more than one thing at a time. 

 This pedagogic principle of concentration 

 is undoubtedly sound. One must, how- 

 ever, learn how to apply it wisely. For 

 instance, in the physical laboratory it is 

 undesirable to introduce experiments which 

 teach the use of the calipers or of the 

 vernier or of the slide rule. Instead of 

 such uninteresting experiments of limited 

 purpose, the students should be directed to 

 extremely interesting problems which in- 

 volve the use of these instruments, and thus 

 be led to learn to use the instruments as a 

 matter of course, and not as a matter of 

 difficulty. Just so the smaller elements of 

 mathematical routine can be made to at- 

 tach themselves to laboratory problems, 

 arousing and retaining the interest of the 



students. Again, everything exists in its 

 relations to other things, and in teaching 

 the one thing the teacher must illuminate 

 these relations. 



Every result of importance should be 

 obtained by at least two distinct methods, 

 and every result of especial importance by 

 two essentially distinct methods. This is 

 possible in mathematics and the physical 

 sciences, and thus the student is made thor- 

 oughly independent of all authority. 



All results should be checked, if only 

 qualitatively or if only 'to the first signifi- 

 cant figure.' In setting pi'oblems in prac- 

 tical mathematics (arithmetical computa- 

 tion or geometrical construction) the teach- 

 er should indicate the amount or percentage 

 of error permitted in the final result. If 

 this amount of percentage is chosen con- 

 veniently in the different examples, the 

 student will be led to the general notion 

 of closer and closer approximation to a 

 perfectly definite result, and thus in a prac- 

 tical way to the fundamental notions of 

 the theory of limits and of irrational num- 

 bers. Thus, for instance, uniformity of 

 convergence can be taught beautifully in 

 connection with the concrete notion of area 

 under a monotonic curve between two or- 

 dinates, by a figure due to Newton, while 

 the interest will be still greater if in the 

 diagram area stands for work done by an 

 engine. 



The teacher should lead up to an impor- 

 tant theorem gradually in such a way that 

 the precise meaning of the statement in 

 question, and further, the practical— i e., 

 computational or graphical or experimental 

 — truth of the theorem is fully appre- 

 ciated; and, furthermore, the importance 

 of the theorem is understood, and, indeed, 

 the desire for the formal proof of the 

 proposition is awakened, before the formal 

 proof itself is developed. Indeed, in most 

 cases, much of the proof should be secured 



