72 



SCIENCE. 



[Vol. XX. No. 496 



2. Independence of the Laws of Motion. 



Maxwell' maintains that " the denial of Newton's first law 

 is in contradiction to the only system of consistent doctrine 

 about space and time which the human mind has been able 

 to form." If this be so, it must be possible to deduce the law 

 from the doctrine of space and time, and it cannot be held 

 to be hypothetical in character. Maxwell's argument is as 

 follows : " If the velocity [of a body freed from the action of 

 force] does not remain constant, let us suppose it to vary. 

 The change of velocity must have a definite direction and 

 magnitude. By the maxim that the same causes will always 

 produce the same effects, this variation must be the same, 

 whatever be the time or place of the experiment. The 

 direction of the change of motion must therefore be de- 

 termined either by the direction of the motion itself or 

 by some direction fixed in the body. Let us, in the first 

 place, suppose the law to be that the velocity diminishes 

 at a certain rate. . . . The velocity referred to in this hy- 

 pothetical law can only be the velocity referred to a point 

 absolutely at rest. For if it is a relative velocity, its direc- 

 tion as well as its magnitude depends on the velocity of the 

 point of reference. . . . Hence the hypothetical law is with- 

 out meaning unless we admit the possibility of defining ab- 

 solute rest and absolute velocity;" 



This argument, which is endorsed by Tait," may be used 

 to prove Newton's law also to be without meaning. For this 

 purpose all that is necessary is to substitute displacement for 

 velocity or motion, wherever these words occur in the above 

 quotation, and changes for diminishes. The argument is 

 thus transformed into one equally good or bad, in favor of 

 the cessation of motion on the cessation of the action of force, 

 as against Newton's law. 



The fallacy — for the argument would thus appear to be 

 fallacious — seems to lie in the incomplete recognition of 

 the relativity of the law of motion under consideration. 

 Thus, when, in the second sentence of the above quotation. 

 Maxwell says: " The change of velocity must have a definite 

 magnitude and direction," he forgets that its magnitude and 

 direction must vary with the point of reference. And the 

 whole argument turns upon this asserted definiteness. 



While the first law must be considered incapable of deduc- 

 tion, its right to formal enunciation among the fundamental 

 hypotheses of dynamics has often been disputed on the ground 

 of its being a particular case of the second law. This must 

 be admitted; and its separate enunciation must therefore be 

 pronounced illogical. 



There is one objection, however, which may perhaps be 

 urged against the omission of the first law, viz., that Max- 

 well" and other authorities hold that this law, "by stating 

 under what circumstances the velocity of a moving body 

 remains constant, supplies us with a method of defining equal 

 intervals" of time. As no such statement is ever made 

 about the second law, it would thus appear that the omission 

 of the first would leave us without a basis for the measure- 

 ment of time. 



This objection, however, is easily met. For, first, the sec- 

 ond law supplies us with more methods of defining equal 

 intervals of time than the first law. In addition to the defi- 

 nition given by the latter, it tells us, for example, that those 

 intervals are equal in which a body acted upon by a constant 

 force undergoes equal changes of velocity. 



^ Matter and Motion, Art. XLI. 



2 Ency. Brit., 9th Ed., Art. Mechanics, § 298. 



3 Matter and Motion, Art. .XLiii 



Second, both laws assume that equal intervals of time 

 have already been defined. So far as power of defining 

 is concerned, therefore, they give us nothing that we 

 did not possess before their enunciation. The only ad- 

 vance in lime-measurement which we owe them is that 

 they show us how to construct time-pieces which will mark 

 off for us the intervals assumed to be equal in their enunci 

 ation. 



Third, the intervals assumed equal in the enunciation of 

 these laws are not known to be equal. What they assume 

 is therefore nothing more than a conventional time scale; 

 and what they give us is nothing more than certain methods 

 of securing accurate copies of this scale. 



And, fourth, both of these laws may be enunciated so as 

 to retain all their dynamical significance, and yet make no 

 reference to the measurement of time, by adopting as the 

 definition of velocity not distance traversed per unit of time, 

 but the distance traversed while the earth (or, better, a cer- 

 tain ideal earth) rotates through a certain angle relatively 

 to the fixed stars. Enunciated in this way these laws assume 

 no definition of equal intervals of time, and can consequently 

 supply us with no such definitions. 



Newton's second law asserts that the acceleration produced 

 in a body by a force is directly proportional to the force and 

 has the same direction ; and as the assertion is without re- 

 striction, the law implies that the effect of the force is the 

 same, whatever the motion of the body may be and whatever 

 other forces may be acting upon it. Many writers regard 

 the latter implied part of the law as being the only hypo- 

 thetical part. They therefore make it the second law of 

 motion and attempt to deduce the former part from it, the 

 argument being that since any number n of equal and co- 

 directional forces will produce in a body an acceleration n 

 times as great as that produced by one, the acceleration pro- 

 duced in a body must be proportional to the force producing 

 it. It is here assumed, however, that n equal forces in the 

 same direction are equivalent to a single force of n times 

 the magnitude. Thus the explicitly asserted portion of 

 Newton's second law cannot be deduced from the implied 

 portion except by the aid of an additional hypothesis; and 

 the law as a whole must therefore be regarded as hypotheti- 

 cal. 



The third law is supposed to have been deduced from the 

 first by Newton himself. Maxwell^ appears to hold this 

 view; Lodge" declares his adhesion to it; and Tait* says the 

 third law "is very closely connected with the first." New- 

 ton's discussion ' of the third law, in which he is supposed 

 to make this deduction, consists of two parts. He first shows 

 by the experiments referred to above, that the law applies 

 to the case of the stresses between bodies pressing against 

 one another; and he then extends it by the aid of the first 

 law to gravitational stresses, and by the aid of further ex- 

 periment to magnetic stresses as well. In this extension he 

 does not say that he is building upon the results of his ex- 

 periments on impact, but it seems obvious that he does so. 

 Maxwell summarizes his argument admirably in the follow- 

 ing words: "If the attraction of any part of the earth, say> 

 a mountain, upon the remainder of the earth, were greater 

 or less than that of the remainder of the earth upon the 

 mountain, there would be a residual force acting upon the 

 system of the earth and the mountain as a whole, which 



■1 Matter and Motion, Art. ltiii. 

 5 Elementary Mechanics (1885), p. 56. 

 « Properties of Matter (1885), p. ]03. 

 ' Prlnctpla : Scholium to Asiomata. 



