208 



SCIENCE 



[N. S. Vol. XLI. No. 1049 



and a' are the corresponding accelerations, 

 then F/F' = a/a! ; that is, the accelerations 

 are proportional to the forces. 



When once this simple principle is thor- 

 oughly grasped, the student finds himself 

 immediately in a i>ositon to attack any of the 

 elementary problems in the dynamics of a 

 particle (in one dimension). For, by this 

 principle, the effect of any force on a given 

 particle can at once be computed if the effect 

 of any one force on that particle is known. 

 In other words the dynamical properties of any 

 given particle of matter are completely deter- 

 mined hy a single physical experiment on thai 

 particle, and the result of such an experiment 

 must be known or assumed with regard to 

 every particle which enters into the discussion 

 of a dynamical problem.^ It is the chief ad- 

 vantage of the equation F/F' = a/a' that by 

 its use the student is led, by the shortest pos- 

 sible route, into direct and vital contact with 

 this central fact of dynamics — ^namely, that 

 different bodies require different amounts of 

 force to give them any specified acceleration. 

 The whole further development of the science 

 is essentially a matter of working out details, 

 and introducing convenient terminology for 

 such derived quantities as mass, momentum, 

 kinetic energy, work, power, etc. 



What then is the objection to the use of this 

 equation ? 



Professor Hoskins expresses his objection 

 as follows: 



An equation which results from comparing the 

 effects of different forces upon the same tody 

 can not, of course, be regarded as a complete ex- 

 pression of the fundamental law of motion; it is 

 equally important to compare the effects of forces 

 acting upon any different todies. This of neces- 

 sity brings in the body constant which most physi- 

 cists call mass. 



In reply to this objection I would say, in 

 the first place, that the question whether a 

 given equation can be regarded as a " eom- 



3 The ' ' standard weight " of a particle is the 

 force required to give the particle the "standard 

 acceleration," 32.1740 feet per second per second; 

 the standard weight of a composite body is defined 

 as the sum of the standard weights of the particles 

 of which it is composed. 



plete expression of the fundamental law of 

 motion " depends simply on whether all the 

 theorems of dynamics can be deduced from 

 this equation, and not on how the equation 

 itself happens to have been derived. In the 

 second place, I quite agree that in order to 

 handle dynamical problems successfully we 

 must indeed be able to discuss the "effect of 

 different forces on different bodies " ; that is, 

 we must be able to determine the inertia, or 

 mass, of each particle under consideration. 

 But so also must we be able to discuss the 

 momentum and kinetic energy of the differ- 

 ent bodies; but that is no reason why a letter 

 denoting mass, or momentum, or kinetic 

 energy, should appear explicitly in the funda- 

 mental equation. From the point of view of 

 scientific economy, the fewer letters that equa- 

 tion contains, the better. The mass concept, 

 like the concept of momentum or kinetic 

 energy, is a derived concept, both historically 

 and practically, and it seems to me a merit of 

 the plan here advocated that on this plan the 

 derivative character of all these quantities is 

 explicitly apparent in the mathematical devel- 

 opment of the equations. 



So much for what may be called the force 

 method of beginning mechanics. 



A second method of developing the whole 

 subject might be to adopt mass instead of force 

 as the fundamental concept — as has been done, 

 for example, by Mach and by Boltzmann. This 

 method seems to me, however, open to three 

 serious objections. 



First, the instrument commonly taken as 

 the fundamental means of measuring mass — 

 namely, the beam-balance — is essentially a grav- 

 itational instrument, depending for its opera- 

 tion on the (established or assumed) equality 

 of the gravitational fields of force at the two 

 ends of the beam; whereas the instrument for 

 measuring forces, at least in a readily ideal- 

 ized form, is a universal instrument, not in 

 any way dependent on locality. For example, 

 if a man should be placed, in imagination, at 

 the " point of zero gravity " between the earth 

 and the moon, it is not at all obvious how he 

 would proceed to measure a given mass with 

 a beam-balance; whereas, if he had a spring 



