lO J. G. MacGEEGOR on THE 



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 certain ideal earth) rotates through a certain angle relatively 

 to the fixed stars. Enunciated in this way these laws assume no definition of equal in- 

 tervals of time, and can consequently supply us with no such definitions. 



Thus the necessities of time measurement would seem to offer no opposition to the 

 demand of logic that the first law of motion should not be given separate enunciation. 



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 with- 

 out restriction, the law implies that the effect of the force is the same, whateA^er other 

 forces may be acting upon the body. Many writers regard the latter implied part of the law, 

 often called Galileo's law or the law of the jshysical independence of forces, as being the 

 only hypothetical part. They therefore make it the second law of motion and attempt to 

 deduce the former part from it, the argument being usually 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 produced in a body must be proportional to the 

 force producing it. 



At first sight Gralileo's law seems to be a more simple hypothesis than Newton's law. 

 To judge of their relative simplicity, however, it must be noted that in deducing the 

 latter from the former, two additional hypotheses are employed, viz., (1) That the direc- 

 tion of the acceleration produced by a force is the same as that of the force, and (2) That n 

 equal forces, acting in the same direction on a particle, produce the same effect as a single 

 force n times as great as any one of them. Thus while Galileo's law follows at once from 

 Newton's, Newton's can be deduced from Galileo's only by the introduction of two addi- 

 tional hypotheses. The superior simplicity of Galileo's law is thus apparent, not real. 



The third law is supposed by some writers 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" Three 

 questions present themselves here : (1) Did Newton really attempt to deduce the third 

 law from the first ? (2) If so, is his argument sound ? (3) If not, is Maxwell's version 

 of his argument sound ? 



That Newton really regarded himself as having deduced the third law from the first 

 is rendered extremely doubtful by the fact that he retained this law as one of his axioms. 

 But it seems clear (though to speak positively would require a more thorough knowledge 

 of his usage of Latin than I possess) that he regarded part of what we now consider to 

 be included in the third law to be capable of deduction. 



That Newton regarded the third law as less general in its applicability as an axiom, 

 than we do, may be gathered from his comments on it. He illustrates it by reference to 

 the finger pressing a stone, a horse hauling a stone by means of a rope, and bodies im- 

 pinging upon one another, — all cases of palpably contact actions. And he concludes his 



' Matter and Motion, Art. Iviii. 



■•' Elementary Mechanics (1885), p. 56. 



^ Properties of Matter (1885), p. 103. 



