244 Prof. J. Gr. MacGreeor on the 



o 



to the first law. But that is quite consistent with a denial of 

 its logical independence*. 



(3) The asserted Deduction of the Third Law of Motion. 



The question of the possibility of deducing the third law 

 from the first seems to me of such importance f that I shall 

 discuss it at some length. 



The common belief that the deduction is possible arises 

 probably from the fact that Newton is supposed to have made 

 it J. I can best examine Newton's argument § by writing it 



* Prof. Minchin is reported in ' Nature ' (vol. xlviii. p. 166) as not 

 admitting the first law to be a particular case of the second, on the ground 

 that " unless force was postulated (the function of the first law), the second 

 became a mere definition, and not a law.'' Obviously, if the first law 

 be regarded merely as a postulate it cannot be a particular case of a law. 

 But Prof. Minchin states that it gives also " the criterion of the presence 

 of force; " and it is this aspect of the law which is held to be a particular 

 case of the second. Prof. Henrici is reported {ibid.) as saying that " in 

 passing from geometry to kiuematics the idea of time presented itself, 

 and the appropriate axiom was contained in Newton's first law." He 

 would thus make the first law a kinematical axiom, though why such 

 an axiom should be expressed in terms of dynamical conceptions is not 

 apparent. Et would follow, however, from this position that the first law 

 ought not to be enunciated among the dynamical axioms, in which case 

 the question of dependence or independence would not arise. 



t Prof. Lod^e now considers it of minor importance. But he has 

 insisted upon the possibility of deducing this law, not only in his book 

 on Mechanics, but also in papers in this Magazine and in ' The Engineer.' 

 Indeed he still considers the deduction so important for the conversion of 

 examination candidates and engineers as to justify us in pretending, as it 

 were, to make it, by means of a non-rigorous proof (p. 10). 



| " 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 that he re- 

 garded 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 

 impinging upon one another,- — all cases of palpably contact-actions. And 

 he concludes his illustrative comments by saying : — " This law holds also 

 in cases of attraction, as will be proved in the following Scholium." The 

 fact that his third law states action and reaction to be equal and opposite 

 but says nothing as to their being in the same straight line, forms corro- 

 borative evidence that he regarded his law as applicable directly to con- 

 tact-actions only. For in such actions it would follow, from the opposition 

 of action and reaction, that they must be in the same straight line. It 

 would thus appear that Newton regarded the application of the third law 

 to attractions as capable of deduction." — My Address, p. 10. 



§ "In attractionibus rem sic breviter ostendo. Corporibus duobus 

 quibusvis A, B se mutuo trahentibus, concipe obstaculum quodvis inter- 

 poni, quo congressus eorum impediatur. Si corpus alterutrum A magis 

 trahitur versus corpus alterum B, qnam illud alterum B in prius A, 



