Chapter 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.

John Bernoulli, in 1717, first perceived the universal import of the principle of virtual movements for all systems; a discovery stated in a letter to Varignon. Finally, Lagrange gives a general demonstration of the principle and founds upon it his whole _Analytical Mechanics_. But this general demonstration is based after all upon Huygens and Torricelli's remarks. Lagrange, as is known, conceives simple pulleys arranged in the directions of the forces of the system, passes a cord through these pulleys, and appends to its free extremity a weight which is a common measure of all the forces of the system. With no difficulty, now, the number of elements of each pulley may be so chosen that the forces in question shall be replaced by them. It is then clear that if the weight at the extremity cannot sink, equilibrium subsists, because heavy bodies cannot of themselves move upwards. If we do not go so far, but wish to abide by Torricelli's idea, we may conceive every individual force of the system replaced by a special weight suspended from a cord passing over a pulley in the direction of the force and attached at its point of application. Equilibrium subsists then when the common centre of gravity of all the weights together cannot sink. The fundamental supposition of this demonstration is plainly the impossibility of a perpetual motion.

Lagrange tried in every way to supply a proof free from extraneous elements and fully satisfactory, but without complete success. Nor were his successors more fortunate.

The whole of mechanics, thus, is based upon an idea, which, though unequivocal, is yet unwonted and not coequal with the other principles and axioms of mechanics. Every student of mechanics, at some stage of his progress, feels the uncomfortableness of this state of affairs; every one wishes it removed; but seldom is the difficulty stated in words. Accordingly, the zealous pupil of the science is highly rejoiced when he reads in a master like Poinsot (_Théorie générale de l'équilibre et du mouvement des systèmes_) the following passage, in which that author is giving his opinion of the _Analytical Mechanics_:

"In the meantime, because our attention in that work was first
wholly engrossed with the consideration of its beautiful
development of mechanics, which seemed to spring complete from a
single formula, we naturally believed that the science was
completed or that it only remained to seek the demonstration of the
principle of virtual velocities. But that quest brought back all
the difficulties that we had overcome by the principle itself. That
law so general, wherein are mingled the vague and unfamiliar ideas
of infinitely small movements and of perturbations of equilibrium,
only grew obscure upon examination; and the work of Lagrange
supplying nothing clearer than the march of analysis, we saw
plainly that the clouds had only appeared lifted from the course of
mechanics because they had, so to speak, been gathered at the very
origin of that science.

"At bottom, a general demonstration of the principle of virtual
velocities would be equivalent to the establishment of the whole of
mechanics upon a different basis: for the demonstration of a law
which embraces a whole science is neither more nor less than the
reduction of that science to another law just as general, but
evident, or at least more simple than the first, and which,
consequently, would render that useless."[50]

According to Poinsot, therefore, a proof of the principle of virtual movements is tantamount to a total rehabilitation of mechanics.

Another circumstance of discomfort to the mathematician is, that in the historical form in which mechanics at present exists, dynamics is founded on statics, whereas it is desirable that in a science which pretends to deductive completeness the more special statical theorems should be deducible from the more general dynamical principles.

In fact, a great master, Gauss, gave expression to this desire in his presentment of the principle of least constraint (Crelle's _Journal für reine und angewandte Mathematik_, Vol. IV, p. 233) in the following words: "Proper as it is that in the gradual development of a science, and in the instruction of individuals, the easy should precede the difficult, the simple the complex, the special the general, yet the mind, when once it has reached a higher point of view, demands the contrary course, in which all statics shall appear simply as a special case of mechanics." Gauss's own principle, now, possesses all the requisites of universality, but its difficulty is that it is not immediately intelligible and that Gauss deduced it with the help of D'Alembert's principle, a procedure which left matters where they were before.

Whence, now, is derived this strange part which the principle of virtual motion plays in mechanics? For the present I shall only make this reply. It would be difficult for me to tell the difference of impression which Lagrange's proof of the principle made on me when I first took it up as a student and when I subsequently resumed it after having made historical researches. It first appeared to me insipid, chiefly on account of the pulleys and the cords which did not fit in with the mathematical view, and whose action I would much rather have discovered from the principle itself than have taken for granted. But now that I have studied the history of the science I cannot imagine a more beautiful demonstration.

In fact, through all mechanics it is this self-same principle of excluded perpetual motion which accomplishes almost all, which displeased Lagrange, but which he still had to employ, at least tacitly, in his own demonstration. If we give this principle its proper place and setting, the paradox is explained.

The principle of excluded perpetual motion is thus no new discovery; it has been the guiding idea, for three hundred years, of all the great inquirers. But the principle cannot properly be _based_ upon mechanical perceptions. For long before the development of mechanics the conviction of its truth existed and even contributed to that development. Its power of conviction, therefore, must have more universal and deeper roots. We shall revert to this point.