284 



NATURE 



[January 17, 1895 



Newton first introduced all four notions. To this D'Alembert's 

 principle gave the analytical method of treating generally con- 

 nected systems. Beyond it all is deduction. Here Hertz in- 

 troduces a discu-ision as to the so-called forces of inerlia. From 

 his discussing the case of a solid subject to centripetal acceler- 

 ation by means of a strinj, the question is much more intricate 

 than if he had taken the case of a body falling freely under 

 gravity, where the force is applied directly by the earth to each 

 point of the body, and not, as in the caseof thestring.distributed 

 to each part by stresses in the solid. Hertz seems to consider 

 that there is some outstanding confusion in applying the prin- 

 ciple of equality of action and reaction, and appears to hold that 

 by this principle the action on the body requires some reaction 

 in the- body whose acceleration is the effect of the force. He 

 dies not seem fully to appreciate that action and reaction are 

 always on different bodies. From his consideration of this, and 

 fnm a general review of our conception of force, he concludes 

 that there is something mysterious about it, that its nature is a 

 problem in physics, like the nature of electricity. We have a 

 quite distinct conception of velocity : why not of force? He 

 concludes that the mystery is not due to our not having enough 

 ideas to associate with the word, but to our trying to put too 

 much into it. These mysteries, however, do not invalidate in 

 any way the deductions that have been made ; they only require 

 lis to set k out a new foundation for our dynamics. He goes on 

 ' 1 criticise this method of filling nature with forces of which, 

 • eing ultimately between molecules, we can have no direct ex- 

 perience K piece of iron on a table is acted on by gravitation, 

 ohesion, repulsion, magnetic, electromagnetic, electric, and 

 chemical forces. Some of these would drag it to pieces if un- 



alanced to a nicety by others. Is this a sound view of nature? 

 I Jan we not get some more attractive one? 

 A second view may be elaborated by making our fundamental 



uantities, space, time, mass, and energy. There is no book 

 in which this view of nature is fully and consistently worked 

 out, at least none that Hertz was acquainted with. He sketches 

 how it might proceed. Besides the postulate of the conserva- 

 tion of energy we require some definition of potential energy 

 and experimental relations connecting it with space, and in 

 addition we have a choice of relations with kinetic energy, of 

 which Hertz suggests the choice of the integral form of Hamil- 

 ton's principle known as that of least action. This is, no doubt, 

 a recondite idea to use as a fundamental postulate, but it only 

 implicitly involves the idea of force, which then comes in 

 merely as a definition. To this method, which certainly has 

 several great advantages, Heriz makes a number of olijeclions. 

 In the first place he objects that it requires the equations of 

 connection to be integral equations, and we know such actions 

 as pure rolling of one hard body on another cannot be so ex- 

 pressed. \Ve must, in order to specify the subsequent motion, 

 know the rate of rotation round the normal axis through the 

 point of contact, and this cannot be specified except in terms of 

 fliflereniials. To such motions we cannot apply the i)roposed 

 prinriple of lea^t action, and yet we can hardly dispute that 

 such rolling is possible in nature. If we treat it as the limits of 

 frictional sliding, we introduce the whole of the difficulties of 

 force, or of the irregular heat actions which have not yet been 

 fully made amenable to accurate dynamical treatment, .\gain, 

 difficulties arise as to the foundation of this method. There is 

 great difficulty in specifying energy itself. Hnwcan it be satis- 

 factorily measured without returning to the first method, and 

 introducing the i<lca of force? .Some have conceived of energy 

 as a sort of substance ; but when we try to form concrete con- 

 ceptions of what is occurring, we get involved in perplexities. 

 The very existence of two forms of energy is a very serious 

 difficulty. Again, it is doubtful whether it can be sound to 

 consider the integral of lea.si action as a/««a'<3w<r»n'a/ principle. 

 It makes the present depend on the future. It sets the problem 

 to nature to make a certain Integral the minimum. 



A good many of Iheseobjections could begot over by making 

 all energy kinetic, which is what Hertz himself practically 

 assumes in his own method. 



This third method begins by auuming only three fundamental 

 quantities, time, space, and mass, and puis aside as non-funda- 

 mental, force and cncrijy. in order to explain how nature 

 works, we already do postulate invisible umlcrlying structures 

 in nature. We postulate these in (he atoms and molecules of 

 matter. Hertz sees in all actions the working of an underlying 

 structure whose masses and motions are producing the effects 

 on matter that we perceive, and what we call force and energy 



NO. I 3 16, VOL. 51] 



are due to the actions of these invisible structures, which he 

 implicitly identifies with the ether. 



We must, however, assume certain connections between the 

 three quantities, time, space, and mass. Between time and mass 

 there is no direct connection. Space and mass. Hertz con- 

 siders, are connected by the existence of a given miiss at each 

 point of space. He cannot mean here to assume a complete 

 plenum, which would make serious difficulties in the way of 

 the working of what he subsequently assumes to be a structure 

 of rigid bodies ; he must include a vanishingly small density at 

 some points, though perhaps he may have had in view the 

 filling of the interstices between his rigid bodies with a fluid. 

 Any way, he goes on to say that some connection is required 

 between all three quantities, and for this purpose he postulates 

 his great fundamental single law of motion, which he considers 

 is an extension to systems of Newton's first law of motion 

 for a single body ; it is that a system, which is unconnected 

 with any others, moves with constant swiftness along one of its 

 straightest paths. " Systema omne liberum perseverare in 

 statu quo quiescendi vel movendi uniformiter in directissimam." 

 In order to understand what Hertz here means by the path of 

 a system, and by its being straight or curved, requires further 

 explanation ; but from this principle, which is capable of 

 analytical representation, and from the assumption that the 

 connections of a sj'Stem are all rigid, he deduces all the funda- 

 mental principles, conservation of areas, momenium, energy, 

 least action, &c. In considering the motion of any part of a 

 system, we find that we may conveniently introduce certain 

 actions of the oilier parts of the system upon it which are 

 measured by forces, which thus come in as mere definitions. 

 He does not seem to inveslig.ite anywhere the question as to 

 the danger of his rigid connections becoming tangled. 

 Analytically a postulate that the points of two different bodies 

 that act on one another are in contact is easily expressed, but it 

 does not follow that when we come to invent actual rigid 

 connections to produce the observed effects, they will do so for 

 any length of time wiihout jamming. It is a seductive theory 

 that gravitation or electrical actions may be due to vortex 

 filaments ending on atoms ; but the tangling of the filaments is 

 a very serious diflSculty that has not been satisfactorily 

 got over. HerIz does not seem to feel Ihis as a serious 

 difficulty, but he does notice an obvious objection that is 

 sure to be raised, namely, that rigidity in ilself postuKates 

 forces. To this he replies that rigidity in itself is merely a 

 matter of definition and of fact. How is our view of the fact 

 that two points are at a constant distance apart, improved by 

 saying that there is a force between them ? As, however, real 

 bodies are only imperfectly rigid. Hertz concedes that it may 

 be that when we learn more about these invisible connections, 

 they may turn out not to be absolutely rigid. It is a matter for 

 further investigation. This very same view might have been 

 urged, and has been urged already with reference to actions 

 like gravity. The law of gravity can be perfectly well de- 

 scribed without any reference to the notion of force. We may 

 say, every clement of matter moves towards every other ele- 

 ment in the universe with an acceleration inversely pro- 

 portional to the square of their distances apart. We can 

 describe the law kinetically, just as Ileriz proposes to describe 

 the law of motion of parts of a rigid body. There is no 

 iicesiily, however convenient it may be, to introduce the notion 

 of force ; the other bodies in the universe .are a sufficient cause 

 for motion of each, without postulating an entity, force. The 

 principal reason for introducing this notion was to account for 

 a body acting where it was not ; force was invented to get 

 over this ; the body produced force, and this force existed where 

 the body did not, and there acted on other bodies. This whole 

 difficulty seems, however, to be partly due to want of distinct 

 ideas connected with the question of where a boily is. We are 

 so accustomed to consider a body as having a definite bound.ary, 

 that we think there is a definite boundary in reality. All we 

 know of the atoms and molecules, however, would lead us to 

 conclude Ih.at round the centre of e.ich there is a very com- 

 plexly structured region which may or may not change 

 abruptly in structure, but which often extends to con- 

 siderable distances from the atom, so that it is practically 

 impossible to stale absolutely where the atom ends and where 

 the empty space begins. With this view of matter there is 

 no serious reason why we may not rightly consider each atom 

 as existing everywhere that it acts, that is, throughout the 

 whole of space, for its action in causing gravitional .accelera- 



