DEFINITIONS OF FORCE 287 



expressly said so, jocularly remarked " Is it a fact that ' matter ' has any power, 

 either innate or acquired, of resisting external influences ? " Mr Spencer says : 

 "And to Prof. Clerk Maxwell's question thus put, the answer of one not having 

 a like mental peculiarity with Prof. Tait, must surely be No." Mr Spencer, not 

 being aware that the passage is Newton's, and not recognising Maxwell's joke, 

 thinks that Maxwell is at variance with the authors of the book ! 



Finally, Mr Spencer attacks me for inconsistency etc. in my lecture on Force 

 (Nature, September 21, 1876). I do not know how often I may have to answer 

 the perfectly groundless charge of having, in that Lecture, given two incompatible 

 definitions of the same term. At any rate, as the subject is much more important 

 than my estimates of Mr Spencer's accuracy or than his estimates of my " mental 

 peculiarities," I may try to give him clear ideas about it, and to show him that there 

 is no inconsistency on the side of the mathematicians, however the idea of force 

 may have been muddled by the metaphysicians. For that purpose I shall avoid 

 all reference to " differentiations " and " integrations " ; either as they are known 

 to the mathematicians, or as they occur in Mr Spencer's " Formula." Of course 

 a single line would suffice, if the differential calculus were employed. 



Take the very simplest case, a stone of mass M, and weight W, let fall. After 

 it has fallen through a height h, and has thus acquired a velocity v, the Conservation 

 of Energy gives the relation 



M V -= Wh. 



v* 

 Here both sides express real things; M is the kinetic energy acquired, Wh the 



work expended in producing it. 



But if we choose to divide both sides of the equation by - (the average velocity 

 during the fall) we have (by a perfectly legitimate operation) 



Mv= Wt, 



where / is the time of falling. This is read : the momentum acquired is the product 

 of the force into the time during which it has acted. Here, although the equation is 

 strictly correct, it is an equation between purely artificial or non-physical quantities, 

 each as unreal as is the product of a quart into an acre. It is often mathematically 

 convenient, but that is all. The introduction of these artificial quantities is, at least 

 largely, due to the strong (but wholly misleading) testimony of the "muscular" sense. 



Each of these modes of expressing the same truth, of course gives its own 

 mode of measuring (and therefore of defining) force. 



The second form of the equation gives 



W- 



t 



Here, therefore, force appears as the time-rate at which momentum changes ; or, if 

 we please, as the time-rate at which momentum is produced by the force. In using 

 this latter phrase we adopt the convenient, and perfectly misleading, anthropomorphism 

 of the mathematicians. This is the gist of a part of Newton's second Law. 



