188 



SCIENCE 



[N. S. Vol. XLYI. No. 1182 



On the old plan, such an equation is merely 

 a set of instructions for the computation of 

 s. If the body has fallen three seconds, the 

 student is expected on the old plan to write 

 s = * X 32 X 3= = 144. 



This process, simple as it appears to the 

 teacher, is not so simple for the student, as it 

 really involves identifying t as the number of 

 seconds the body has fallen, g as the number of 

 ft./sec.^ in the gravity acceleration, perform- 

 ing the computation and then interpreting the 

 result as a number of feet. One obvious cause 

 of trouble is the necessity for using certain 

 definite units on each side, with the errors 

 made by the use of the wrong units; and 

 another, perhaps not so obvious, is the fact 

 that the formula itself is not a statement 

 ahout a real distance of so many feet, a real 

 acceleration of so many ft./sec.° and a real 

 time of so many seconds, but about puo-e num- 

 bers, mere incomplete " so many s," the most 

 abstract things yet invented by man. Under 

 these conditions is it surprising that a fresh- 

 man fails to formulate his data into mathe- 

 matical equations? 



On the Hew plan, the equation is taken as 

 a statement about actual concrete things. In 

 this particular case, the computation would 

 take the form, 



s = * X 32 ™ X 32 sec.2 = 144 ft. 

 sec- 



The interpretation of the formula is now that 

 s is physically a result of the combination of 

 the gravity acceleration g with the time t, 

 which enters once in producing the final veloc- 

 ity gt, and mean velocity igt and again in 

 combination with this mean velocity to give 

 the distance igP. The essential feature in 

 the application of this plan is the insertion of 

 each quantity as a quantity, that is, as so 

 many times another quantity of the same hind, 

 and not as a mere " so many." 



If in computation the boy should happen 

 to forget to square t, he would get 



calling i^ -is min., he 

 s = i X 32 



s = i X 32 



X 3 sec. 



sec' 



ft. J^ • 2 _ 1 ft. min.'' 

 gjjgl X 2Q-2 nun. - ^5 ^^^2 ■ 



To reduce this to simpler terms he has only 

 to substitute 60- sec- for min^, exactly as he 

 would perform any other algebraic substitu- 

 tion of equals, and then cancel the sec^ and 

 finish the computation. Or, if he lets 



hr. sec. 

 he gets 



s = § X 22 r^?'^ X 3= sec.^ 

 hr. sec. 



= *X22 3g^;5^,X3^sec.^ = ,V,mm., 



which is as correct an answer as the other. 

 To reduce units the game is simply to substi- 

 tute equals for equals and cancel. If this does 

 not give the right kind of an answer, it is a 

 sure indication of an error. 



Of course, to play the game fairly, we must 

 abolish formulas with lost units, such as s = 

 16i". Examples of these are found most fre- 

 quently in electricity. The old plan would 

 write such a formula as that for the force on 

 a wire in a magnetic field, as F = IIH with 

 a string of restrictions on units, ot F = jqHS 

 with another string. By forgetting the re- 

 strictions and using the simpler formula with 

 the most familiar units, the students often 

 achieve remarkable results. On the new plan 

 this would be written F = KIIH where 



K 



dvne 



an obviously impossible kind of answer. But 

 if he departs from the above method only in 



amp. cm. gauss. 



and all restrictions are removed. It is of 

 of course true that this form of the equation 

 involves more writing than the others; indeed, 

 it may be noted here that the process of treat- 

 ing all equations as physical statements is 

 not necessarily worth while for trained men 

 doing routine computations, but it is extremely 

 useful for all sorts of cases where the com- 

 putations are not familiar enough to be clas- 

 sified as routine work. For all such cases it 

 is well worth while to write out the propor- 

 tionality constant, especially if some one is 

 likely to want I, say, in inches or F in kilo- 

 grams. 



