Part 9

[Footnote 33: A sort of agreement exists between the notions of thermal and electrical capacity, but the difference between the two ideas also should be carefully borne in mind. The thermal capacity of a body depends solely upon that body itself. The electrical capacity of a body _K_ is influenced by all bodies in its vicinity, inasmuch as the charge of these bodies is able to alter the potential of _K_. To give, therefore, an unequivocal significance to the notion of the capacity (_C_) of a body _K_, _C_ is defined as the relation _Q_/_V_ for the body _K_ in a certain given position of all neighboring bodies, and during connexion of all neighboring conductors with the earth. In practice the situation is much simpler. The capacity, for example, of a jar, the inner coating of which is almost enveloped by its outer coating, communicating with the ground, is not sensibly affected by charged or uncharged adjacent conductors.]

[Footnote 34: These formulæ easily follow from Newton's theorem that a homogeneous spherical shell, whose elements obey the law of the inverse squares, exerts no force whatever on points within it but acts on points without as if the whole mass were concentrated at its centre. The formulæ next adduced also flow from this proposition.]

[Footnote 35: The energy of a sphere of radius _r_ charged with the quantity _q_ is 1/2(_q_²/_r_). If the radius increase by the space _dr_ a loss of energy occurs, and the work done is 1/2(_q_²/_r_²)_dr_. Letting _p_ denote the uniform electrical pressure on unit of surface of the sphere, the work done is also 4_r_²[pi]_pdr_. Hence _p = (1/8r²[pi])(q²/r²)_. Subjected to the same superficial pressure on all sides, say in a fluid, our half sphere would be an equilibrium. Hence we must make the pressure _p_ act on the surface of the great circle to obtain the effect on the balance, which is _r²[pi]p = 1/8(q²/r²) = 1/8V²_.]

[Footnote 36: The arrangement described is for several reasons not fitted for the actual measurement of potential. Thomson's absolute electrometer is based upon an ingenious modification of the electrical balance of Harris and Volta. Of two large plane parallel plates, one communicates with the earth, while the other is brought to the potential to be measured. A small movable superficial portion _f_ of this last hangs from the balance for the determination of the attraction _P_. The distance of the plates from each other being _D_ we get _V = D[sqrt](8[pi]P/f)_.]

[Footnote 37: This moment of torsion needs a supplementary correction, on account of the vertical electric attraction of the excited disks. This is done by changing the weight of the disk by means of additional weights and by making a second reading of the angles of deflexion.]

[Footnote 38: The jar in our experiment acts like an accumulator, being charged by a dynamo machine. The relation which obtains between the expended and the available work may be gathered from the following simple exposition. A Holtz machine _H_ (Fig. 40) is charging a unit jar _L_, which after _n_ discharges of quantity _q_ and potential _v_, charges the jar _F_ with the quantity _Q_ at the potential _V_. The energy of the unit-jar discharges is lost and that of the jar _F_ alone is left. Hence the ratio of the available work to the total work expended is

_½QV/[½QV + (n/2)qv]_ and as _Q = nq_, also _V/(V + v)_.

If, now, we interpose no unit jar, still the parts of the machine and the wires of conduction are themselves virtually such unit jars and the formula still subsists _V/(V + [sum]v)_, in which [sum]_v_ represents the sum of all the successively introduced differences of potential in the circuit of connexion.]

ON THE PRINCIPLE OF THE CONSERVATION OF ENERGY.[39]

In a popular lecture, distinguished for its charming simplicity and clearness, which Joule delivered in the year 1847,[40] that famous physicist declares that the living force which a heavy body has acquired by its descent through a certain height and which it carries with it in the form of the velocity with which it is impressed, is the _equivalent_ of the attraction of gravity through the space fallen through, and that it would be "absurd" to assume that this living force could be destroyed without some restitution of that equivalent. He then adds: "You will therefore be surprised to hear that until very _recently_ the universal opinion has been that living force could be absolutely and irrevocably destroyed at any one's option." Let us add that to-day, after forty-seven years, the _law of the conservation of energy_, wherever civilisation exists, is accepted as a fully established truth and receives the widest applications in all domains of natural science.

The fate of all momentous discoveries is similar. On their first appearance they are regarded by the majority of men as errors. J. R. Mayer's work on the principle of energy (1842) was rejected by the first physical journal of Germany; Helmholtz's treatise (1847) met with no better success; and even Joule, to judge from an intimation of Playfair, seems to have encountered difficulties with his first publication (1843). Gradually, however, people are led to see that the new view was long prepared for and ready for enunciation, only that a few favored minds had perceived it much earlier than the rest, and in this way the opposition of the majority is overcome. With proofs of the fruitfulness of the new view, with its success, confidence in it increases. The majority of the men who employ it cannot enter into a deep-going analysis of it; for them, its success is its proof. It can thus happen that a view which has led to the greatest discoveries, like Black's theory of caloric, in a subsequent period in a province where it does not apply may actually become an obstacle to progress by its blinding our eyes to facts which do not fit in with our favorite conceptions. If a theory is to be protected from this dubious rôle, the grounds and motives of its evolution and existence must be examined from time to time with the utmost care.

The most multifarious physical changes, thermal, electrical, chemical, and so forth, can be brought about by mechanical work. When such alterations are reversed they yield anew the mechanical work in exactly the quantity which was required for the production of the part reversed. This is the _principle of the conservation of energy_; "energy" being the term which has gradually come into use for that "indestructible something" of which the measure is mechanical _work_.

How did we acquire this idea? What are the sources from which we have drawn it? This question is not only of interest in itself, but also for the important reason above touched upon. The opinions which are held concerning the foundations of the law of energy still diverge very widely from one another. Many trace the principle to the impossibility of a perpetual motion, which they regard either as sufficiently proved by experience, or as self-evident. In the province of pure mechanics the impossibility of a perpetual motion, or the continuous production of _work_ without some _permanent_ alteration, is easily demonstrated. Accordingly, if we start from the theory that all physical processes are purely _mechanical_ processes, motions of molecules and atoms, we embrace also, by this _mechanical_ conception of physics, the impossibility of a perpetual motion in the _whole_ physical domain. At present this view probably counts the most adherents. Other inquirers, however, are for accepting only a purely _experimental_ establishment of the law of energy.

It will appear, from the discussion to follow, that _all_ the factors mentioned have co-operated in the development of the view in question; but that in addition to them a logical and purely formal factor, hitherto little considered, has also played a very important part.

I. THE PRINCIPLE OF THE EXCLUDED PERPETUAL MOTION.

The law of energy in its modern form is not identical with the principle of the excluded perpetual motion, but it is very closely related to it. The latter principle, however, is by no means new, for in the province of mechanics it has controlled for centuries the thoughts and investigations of the greatest thinkers. Let us convince ourselves of this by the study of a few historical examples.

S. Stevinus, in his famous work _Hypomnemata mathematica_, Tom. IV, _De statica_, (Leyden, 1605, p. 34), treats of the equilibrium of bodies on inclined planes.

Over a triangular prism _ABC_, one side of which, _AC_, is horizontal, an endless cord or chain is slung, to which at equal distances apart fourteen balls of equal weight are attached, as represented in cross-section in Figure 41. Since we can imagine the lower symmetrical part of the cord _ABC_ taken away, Stevinus concludes that the four balls on _AB_ hold in equilibrium the two balls on _BC_. For if the equilibrium were for a moment disturbed, it could never subsist: the cord would keep moving round forever in the same direction,--we should have a perpetual motion. He says:

"But if this took place, our row or ring of balls would come once more into their original position, and from the same cause the eight globes to the left would again be heavier than the six to the right, and therefore those eight would sink a second time and these six rise, and all the globes would keep up, of themselves, _a continuous and unending motion, which is false_."[41]

Stevinus, now, easily derives from this principle the laws of equilibrium on the inclined plane and numerous other fruitful consequences.

In the chapter "Hydrostatics" of the same work, page 114, Stevinus sets up the following principle: "Aquam datam, datum sibi intra aquam locum servare,"--a given mass of water preserves within water its given place.

This principle is demonstrated as follows (see Fig. 42):

"For, assuming it to be possible by natural means, let us suppose that A does not preserve the place assigned to it, but sinks down to D. This being posited, the water which succeeds A will, for the same reason, also flow down to _D_; _A_ will be forced out of its place in _D_; and thus this body of water, for the conditions in it are everywhere the same, _will set up a perpetual motion, which is absurd_."[42]

From this all the principles of hydrostatics are deduced. On this occasion Stevinus also first develops the thought so fruitful for modern analytical mechanics that the equilibrium of a system is not destroyed by the addition of rigid connexions. As we know, the principle of the conservation of the centre of gravity is now sometimes deduced from D'Alembert's principle with the help of that remark. If we were to reproduce Stevinus's demonstration to-day, we should have to change it slightly. We find no difficulty in imagining the cord on the prism possessed of unending uniform motion if all hindrances are thought away, but we should protest against the assumption of an accelerated motion or even against that of a uniform motion, if the resistances were not removed. Moreover, for greater precision of proof, the string of balls might be replaced by a heavy homogeneous cord of infinite flexibility. But all this does not affect in the least the historical value of Stevinus's thoughts. It is a fact, Stevinus deduces apparently much simpler truths from the principle of an impossible perpetual motion.

In the process of thought which conducted Galileo to his discoveries at the end of the sixteenth century, the following principle plays an important part, that a body in virtue of the velocity acquired in its descent can rise exactly as high as it fell. This principle, which appears frequently and with much clearness in Galileo's thought, is simply another form of the principle of excluded perpetual motion, as we shall see it is also in Huygens.

Galileo, as we know, arrived at the law of uniformly accelerated motion by _a priori_ considerations, as that law which was the "simplest and most natural," after having first assumed a different law which he was compelled to reject. To verify his law he executed experiments with falling bodies on inclined planes, measuring the times of descent by the weights of the water which flowed out of a small orifice in a large vessel. In this experiment he assumes as a fundamental principle, that the velocity acquired in descent down an inclined plane always corresponds to the vertical height descended through, a conclusion which for him is the immediate outcome of the fact that a body which has fallen down one inclined plane can, with the velocity it has acquired, rise on another plane of any inclination only to the same vertical height. This principle of the height of ascent also led him, as it seems, to the law of inertia. Let us hear his own masterful words in the _Dialogo terzo_ (_Opere_, Padova, 1744, Tom. III). On page 96 we read:

"I take it for granted that the velocities acquired by a body in descent down planes of different inclinations are equal if the heights of those planes are equal."[43]

Then he makes Salviati say in the dialogue:[44]

"What you say seems very probable, but I wish to go further and by an experiment so to increase the probability of it that it shall amount almost to absolute demonstration. Suppose this sheet of paper to be a vertical wall, and from a nail driven in it a ball of lead weighing two or three ounces to hang by a very fine thread _AB_ four or five feet long. (Fig. 43.) On the wall mark a horizontal line _DC_ perpendicular to the vertical _AB_, which latter ought to hang about two inches from the wall. If now the thread _AB_ with the ball attached take the position _AC_ and the ball be let go, you will see the ball first descend through the arc _CB_ and passing beyond _B_ rise through the arc _BD_ almost to the level of the line _CD_, being prevented from reaching it exactly by the resistance of the air and of the thread. From this we may truly conclude that its impetus at the point _B_, acquired by its descent through the arc _CB_, is sufficient to urge it through a similar arc _BD_ to the same height. Having performed this experiment and repeated it several times, let us drive in the wall, in the projection of the vertical _AB_, as at _E_ or at _F_, a nail five or six inches long, so that the thread _AC_, carrying as before the ball through the arc _CB_, at the moment it reaches the position _AB_, shall strike the nail _E_, and the ball be thus compelled to move up the arc _BG_ described about _E_ as centre. Then we shall see what the same impetus will here accomplish, acquired now as before at the same point _B_, which then drove the same moving body through the arc _BD_ to the height of the horizontal _CD_. Now gentlemen, you will be pleased to see the ball rise to the horizontal line at the point _G_, and the same thing also happen if the nail be placed lower as at _F_, in which case the ball would describe the arc _BJ_, always terminating its ascent precisely at the line _CD_. If the nail be placed so low that the length of thread below it does not reach to the height of _CD_ (which would happen if _F_ were nearer _B_ than to the intersection of _AB_ with the horizontal _CD_), then the thread will wind itself about the nail. This experiment leaves no room for doubt as to the truth of the supposition. For as the two arcs _CB_, _DB_ are equal and similarly situated, the momentum acquired in the descent of the arc _CB_ is the same as that acquired in the descent of the arc _DB_; but the momentum acquired at _B_ by the descent through the arc _CB_ is capable of driving up the same moving body through the arc _BD_; hence also the momentum acquired in the descent _DB_ is equal to that which drives the same moving body through the same arc from _B_ to _D_, so that in general every momentum acquired in the descent of an arc is equal to that which causes the same moving body to ascend through the same arc; but all the momenta which cause the ascent of all the arcs _BD_, _BG_, _BJ_, are equal since they are made by the same momentum acquired in the descent _CB_, as the experiment shows: therefore all the momenta acquired in the descent of the arcs _DB_, _GB_, _JB_ are equal."

The remark relative to the pendulum may be applied to the inclined plane and leads to the law of inertia. We read on page 124:[45]

"It is plain now that a movable body, starting from rest at _A_ and descending down the inclined plane _AB_, acquires a velocity proportional to the increment of its time: the velocity possessed at _B_ is the greatest of the velocities acquired, and by its nature immutably impressed, provided all causes of new acceleration or retardation are taken away: I say acceleration, having in view its possible further progress along the plane extended; retardation, in view of the possibility of its being reversed and made to mount the ascending plane _BC_. But in the horizontal plane _GH_ its equable motion, according to its velocity as acquired in the descent from _A_ to _B_, will be continued _ad infinitum_." (Fig. 44.)

Huygens, upon whose shoulders the mantel of Galileo fell, forms a sharper conception of the law of inertia and generalises the principle respecting the heights of ascent which was so fruitful in Galileo's hands. He employs the latter principle in the solution of the problem of the centre of oscillation and is perfectly clear in the statement that the principle respecting the heights of ascent is identical with the principle of the excluded perpetual motion.

The following important passages then occur (Hugenii, _Horologium oscillatorium, pars secunda_). _Hypotheses_:

"If gravity did not exist, nor the atmosphere obstruct the motions of bodies, a body would keep up forever the motion once impressed upon it, with equable velocity, in a straight line."[46]

In part four of the _Horologium de centro oscillationis_ we read:

"If any number of weights be set in motion by the force of gravity, the common centre of gravity of the weights as a whole cannot possibly rise higher than the place which it occupied when the motion began.

"That this hypothesis of ours may arouse no scruples, we will state that it simply imports, what no one has ever denied, that heavy bodies do not move _upwards_.--And truly if the devisers of the new machines who make such futile attempts to construct a perpetual motion would acquaint themselves with this principle, they could easily be brought to see their errors and to understand that the thing is utterly impossible by mechanical means."[47]

There is possibly a Jesuitical mental reservation contained in the words "mechanical means." One might be led to believe from them that Huygens held a non-mechanical perpetual motion for possible.

The generalisation of Galileo's principle is still more clearly put in Prop. IV of the same chapter:

"If a pendulum, composed of several weights, set in motion from rest, complete any part of its full oscillation, and from that point onwards, the individual weights, with their common connexions dissolved, change their acquired velocities upwards and ascend as far as they can, the common centre of gravity of all will be carried up to the same altitude with that which it occupied before the beginning of the oscillation."[48]

On this last principle now, which is a generalisation, applied to a system of masses, of one of Galileo's ideas respecting a single mass and which from Huygens's explanation we recognise as the principle of excluded perpetual motion, Huygens grounds his theory of the centre of oscillation. Lagrange characterises this principle as precarious and is rejoiced at James Bernoulli's successful attempt, in 1681, to reduce the theory of the centre of oscillation to the laws of the lever, which appeared to him clearer. All the great inquirers of the seventeenth and eighteenth centuries broke a lance on this problem, and it led ultimately, in conjunction with the principle of virtual velocities, to the principle enunciated by D'Alembert in 1743 in his _Traité de dynamique_, though previously employed in a somewhat different form by Euler and Hermann.

Furthermore, the Huygenian principle respecting the heights of ascent became the foundation of the "law of the conservation of living force," as that was enunciated by John and Daniel Bernoulli and employed with such signal success by the latter in his _Hydrodynamics_. The theorems of the Bernoullis differ in form only from Lagrange's expression in the _Analytical Mechanics_.

The manner in which Torricelli reached his famous law of efflux for liquids leads again to our principle. Torricelli assumed that the liquid which flows out of the basal orifice of a vessel cannot by its velocity of efflux ascend to a greater height than its level in the vessel.

Let us next consider a point which belongs to pure mechanics, the history of the principle of _virtual motions_ or _virtual velocities_. This principle was not first enunciated, as is usually stated, and as Lagrange also asserts, by Galileo, but earlier, by Stevinus. In his _Trochleostatica_ of the above-cited work, page 72, he says:

"Observe that this axiom of statics holds good here:

"As the space of the body acting is to the space of the body acted upon, so is the power of the body acted upon to the power of the body acting."[49]

Galileo, as we know, recognised the truth of the principle in the consideration of the simple machines, and also deduced the laws of the equilibrium of liquids from it.

Torricelli carries the principle back to the properties of the centre of gravity. The condition controlling equilibrium in a simple machine, in which power and load are represented by weights, is that the common centre of gravity of the weights shall not sink. Conversely, if the centre of gravity cannot sink equilibrium obtains, because heavy bodies of themselves do not move upwards. In this form the principle of virtual velocities is identical with Huygens's principle of the impossibility of a perpetual motion.