188 



SCIENCE 



[N. S. Vol. XLV. No. 1156 



Vector algebra is the science of directed 

 lines, or displacements, in space as ordinary 

 algebra is the science of numbers. We do not 

 hesitate to apply the laws of ordinary algebra 

 to quantities which, can be represented by 

 numbers, we need no more have any compimc- 

 tion about applying the laws of vectors to 

 quantities which can be represented by di- 

 rected lines. The expression " the law of par- 

 allelogram of forces " is a provincialism for 

 which there is about as little justification as 

 there would be for a "law of the addition of 

 apples " in arithmetic. The addition law of 

 arithmetic is a law of numbers and is not 

 peculiar to apples; it can be applied to apples 

 not because they have certain desirable prop- 

 erties, but because they can be counted. Sim- 

 ilarly, the parallelogram law is a law of dis- 

 placements or directed lines, and not at all 

 characteristic of forces, but it can be applied 

 to forces because these have among other phys- 

 ical properties those of direction and of mag- 

 nitude and consequently may be represented 

 by directed lines. 



This is the precise point of view which I 

 have adopted in my book toward the directed 

 magnitudes of mechanics. After giving a 

 clear and concise exposition of the laws of addi- 

 tion and resolution of vectors in the first 

 chapter I have applied them to directed quan- 

 tities without hesitation. This mode of pro- 

 cedure is not only correct, but it is straight- 

 forward and simple, as the reviewer admits 

 when he says : 



If the author is correct . . . then certainly the 

 theory underlyiiig the composition and resolution 

 of directed quantities becomes very simple. 



The second question which Professor Eettger 

 raises has to do with my formulation of the 

 principle underlying the science of dynamics. 

 In my book I have based dynamics upon the 

 following principle, which I have called the 

 action principle: 



The vector sum of all the external actions to 

 which a system of particles or any part of it is 

 subject at any instant vanishes. 



SA=0. 

 A particle may be acted upon by other par- 

 ticles and by the ether. The action of one 



particle upon another particle is known as a 

 force. The action of the ether upon a particle 

 I have called a hinetic reaction. Therefore 

 the action principle states 



2(I'-t-q)=0, 



where F denotes a force and q a kinetic re- 

 action. The kinetic reaction on a particle is 

 oppositely directed from and proportional to 

 the acceleration and the constant of propor- 

 tionality is the characteristic constant of the 

 particle known as mass. Therefore we have 

 2(F — m a) =0. 



Commenting upon this principle. Professor 

 Rettger says : 



The reviewer does not wish to say that the au- 

 thor is wrong in his conception. All he wishes to 

 eay is that he entirely fails to appreciate the au- 

 thor's point of view. 



This lack of appreciation is due, it seems 

 to me, to a lack of clear understanding, indi- 

 cated by the following questions, of the nature 

 and function of the kinetic reaction. 



Why is it that the ether acts on a body only 

 when it is being accelerated and not when the 

 body is moving with constant velocity? 



If kinetic reaction is the action of the ether on a 

 particle, and if it is the same kind of a quantity 

 as force (is a force in fact), and if the resultant 

 force F acting on a particle and the kinetic reac- 

 tion g are always equal in magnitude but opposite 

 in direction (both equal to ma in magnitude), why 

 is the body not ia equilibrium? 



If in the first of these questions the term 

 " why " is used in the metaphysical sense, there 

 is no answer for it, except possibly the equally 

 metaphysical answer " because." On the 

 other hand, if it is used to mean " how is this 

 fact correlated with other facts ? " I would 

 state that the answer belongs to electrodynam- 

 ics and not to mechanics and would refer the 

 reviewer to a modern treatise on electro- 

 dynamics, Lorentz's book on " Electron 

 Theory," for instance, where the question is 

 answered at length. 



Answering the second question, one might 

 state: " The body is not in equilibrium for the 

 same reason that a particle revolving in a 

 circle is not in static equilibrium in spite of 



