December 4, 1914] 



SCIENCE 



819 



present writer this procedure is not strictly 

 sound as a scientific explanation of the equa- 

 tion F = Kma. The presence of the mass 

 constant in the equation should rather be ac- 

 cepted as an ultimate part of the laws of mo- 

 tion, while the facts of gravity are whoUy 

 apart from those laws. This is in fact else- 

 where recognized by the author in his appar- 

 ent acceptance of Newton's laws (p. 155) as 

 the scientific basis of dynamics. 



Although Professor Maurer's procedure de- 

 scribed above seems to lend some countenance 

 to the position of those who call the equation 

 F/F' = a/a' the "fundamental equation of 

 dynamics," it is not likely that he really accepts 

 this view; probably his order of presentation 

 was dictated by pedagogic considerations. An 

 equation which results from comparing the 

 effects of different forces upon the same body 

 can not, of course, be regarded as a complete 

 expression of the fundamental law of motion; 

 it is equally important to compare the effects 

 of forces acting upon any different bodies. 

 This of necessity brings in the body-constant 

 which most physicists call mass. If an equa- 

 tion is used which does not contain this quan- 

 tity explicitly, it must be implicitly taken ac- 

 count of in the application. 



As a matter of fact it is difficult to under- 

 stand the antagonism which some critics have 

 shown for the equation F = ma. The main 

 alleged objection to it appears to be the fact 

 that it requires units to be properly chosen ; but 

 this is true of most of the equations used to 

 express physical laws or facts.^ One who 

 really understands the fundamental principle 

 of dynamics will have no difficulty in under- 

 standing the equation F = m,a, or in remember- 

 ing that it implies that units are so defined 

 that unit force acting on unit mass causes unit 

 acceleration. 



To the present writer it seems that the real 



2 It is true, for example, of the equation usually 

 employed to express the law of gravitation. It is 

 true also of the simple equation A = U, where A 

 is the area of a square of side i. In fact it is not 

 easy to cite equations practically used in applied 

 mathematics of which it is not true. The only way 

 to avoid the alleged objection is to throw every 

 such equation into the form of a proportion. 



meaning of the fundamental equation of 

 dynamics is most clearly brought out by pre- 

 senting it first in the form of a compound 

 proportion, 



a'~ F'' m' 



in which a is the acceleration due to a force 

 F acting on a mass m and a' the acceleration 

 due to a force F' acting on a mass to'. From 

 this it is easy to pass to the equation F = Ema 

 for any arbitrary set of units, and then by a 

 certain choice of units to the equation most 

 commonly used because simplest, F = ma. Sub- 

 stantially this method of presentation was 

 given in the first edition of Professor Maurer's 

 book, but seems to have been omitted from the 

 present edition. 



As regards units of force. Professor Maurer's 

 practise is the usual one among engineers. For 

 ordinary use the pound-force or kilogram-force 

 is adopted as unit, with the explanation that 

 although this unit varies with locality, the 

 variation is so slight as ordinarily to be of no 

 practical importance. Although remarking 

 that the pound-force can be made definite by 

 specifying a standard locality, the author does 

 not urge the general adoption of such a stand- 

 ard unit, but follows common scientific usage 

 in adopting a kinetic definition of the abso- 

 lute unit of force. This definition is, of 

 coujse, based upon the fundamental principle 

 of which the equation F = mo (or F = Kma) 

 is an expression. This principle being as- 

 sumed, it is possible to define the unit force 

 as that force which would give some definite 

 mass some definite acceleration; the common 

 practise being to take as unit the force which 

 gives the (arbitrarily chosen) unit mass the 

 (arbitrarily chosen) unit acceleration,^ thus 



3 It is worthy of remark that the advocates of 

 the adoption of a standard pound force, though 

 ostensibly defining it as the weight of a pound body 

 at a standard locality, in reality define it as the 

 force which would give a pound body (meaning a 

 body whose mass is a pound) the acceleration 

 32.1740 ft./s6c=. Thus the real definition is of the 

 same kind as that of the dyne, and the standard 

 pound-force is really a "kinetic" rather than a 

 "gravity" unit. 



