October 15, 1915] 



SCIENCE 



529 



zero, and impact as a special case in which 

 large forces act through small periods of time. 

 This, however, is a distinctly recent move- 

 ment. The older method of procedure was to 

 study, first, statics and problems in impact and 

 thereupon to proceed to kinetics. The reason for 

 this order was probably not wholly logical but 

 largely pedagogic or historical. A student who 

 has a small knowledge of trigonometry is quite 

 fitted, mathematically, to study both statics 

 and problems in impact; whereas, to obtain 

 valuable training in kinetics a knowledge of 

 the differential and integral calculus, includ- 

 ing the simpler differential equations, is nec- 

 essary. Moreover, as a matter of history, 

 statics and impact precedes, I believe, kinet- 

 ics. Let us suppose that the student has fol- 

 lowed this historic and pedagogical order. In 

 his statics he will have learned to deal with 

 forces; these forces may be measured in any 

 units that are convenient, provided only that 

 all the forces are measured in the same units 

 in the same equation; for the equations of 

 statics are homogeneous in the forces. (I, of 

 course, am speaking only of elementary statics, 

 not of the theory of virtual velocities or of po- 

 tential energy.) In studying impact the fun- 

 damental conception is that of momentum. 

 The student learns that momentum is the 

 product of mass by velocity; that momentum 

 is resoluble as are forces; and that in impact 

 the momentum of a system is conserved. He 

 is then in a position to solve problems in in- 

 elastic impact of particles and, with an addi- 

 tional simple law concerning relative veloci- 

 ties, he can proceed to elastic impact. In the 

 problems in impact the units of mass may also 

 be anything, provided, again, that they are the 

 same for all masses; for here again the equa- 

 tions are homogeneous in the masses. 



When, now, such a student comes to kinet- 

 ics he is able at once to proceed to Newton's 

 second law, namely, that the rate of change of 

 momentum is equal to the force. Here, how- 

 ever, we have an equation which is no longer 

 homogeneous either in the mass or in the 

 force, and it is evident, or can be made so to 

 any student, that he can not use arbitrary units 

 of mass and force, but that the two units must 



be in some way correlated. Indeed we should 

 state the second law in the Newtonian form: 

 The rate of change of momentum is propor- 

 tional to, or varies as, the force. We then 

 write 



— {mv) =Jcf. 



The constant h, like any factor of proportion- 

 ality, is determined by substituting the known 

 values for some special case. We naturally select 

 the simplest; that is, a mass falling under its 

 own weight. If now we measure mass in pounds, 

 as we (probably) did in the theory of impact, 

 and force likewise in pounds, as we (likely) did 

 in statics, we find that the mass of weight W 

 has, under the force of weight W, an accelera- 

 tion g; hence 



di'' 



-(Wv)^Wg = 'kW 



or h^g. We therefore have the fundamental 

 equation of kinetics in the form 



~{rnv)=gf. 



If we desire to use some other system of units 

 for mass and force we should likewise have to 

 determine a constant h. 



It is, of course, true that a weight is not a 

 definite constant thing from place to place, but 

 I should not think of calling the student's at- 

 tention very vigorously to this difficulty at 

 this stage, particularly as it again is no diffi- 

 culty at all, provided mass and force are both 

 measured in weights at the same place. Nor 

 do we need to mention that the equation which 

 involves the momentum is one which can still 

 be regarded as valid when the student reaches 

 the theory of relativity and modern electro- 

 dynamics whereas the equation ma^f or any 

 equation involving accelerations leads to the 

 ridiculously needless concepts of transverse 

 and longitudinal (and an infinity of oblique) 

 masses. 



It has always seemed to me that the historic 

 and pedagogical method of procedure was still 

 the best, notwithstanding the above mentioned 

 and modern style. It is quite true that from 

 a logical point of view things proceed more 

 simply when we start with kinetics; but logic 



