been a _fundamental concept_ of mechanics, as also a very important
notion in the applied sciences. The transformation of work into living
force, and of living force into work, suggests directly the notion of
energy--the idea having been first fruitfully employed by Huygens,
although Thomas Young first called it by the _name_ of "energy." Let us
add to this the constancy of weight (really the constancy of mass) and
we shall see that with respect to mechanical energy it is involved in
the very definition of the term that the capacity for work or the
potential energy of a weight is proportional to the height of the level
at which it is, in the geometrical sense, and that it decreases on the
lowering of the weight, on transformation, proportionally to the height
of the level. The zero level here is wholly arbitrary. With this,
equation (2) is given, from which all the other forms follow.
When we reflect on the tremendous start which mechanics had over the
other branches of physics, it is not to be wondered at that the attempt
was always made to apply the notions of that science wherever this was
possible. Thus the notion of mass, for example, was imitated by Coulomb
in the notion of quantity of electricity. In the further development of
the theory of electricity, the notion of work was likewise immediately
introduced in the theory of potential, and heights of electrical level
were measured by the work of unit of quantity raised to that level. But
with this the preceding equation with all its consequences is given for
electrical energy. The case with the other energies was similar.
_Thermal_ energy, however, appears as a special case. Only by the
peculiar experiments mentioned could it be discovered that heat is an
energy. But the measure of this energy by Black's quantity of heat is
the outcome of fortuitous circumstances. In the first place, the
accidental slight variability of the capacity for heat _c_ with the
temperature, and the accidental slight deviation of the usual
thermometrical scales from the scale derived from _the tensions of
gases_, brings it about that the notion "quantity of heat" can be set up
and that the quantity of heat _ct_ corresponding to a difference of
temperature _t_ is nearly proportional to the energy of the heat. It is
a quite accidental historical circumstance that Amontons hit upon the
idea of measuring temperature by the tension of a gas. It is certain in
this that he did not think of the work of the heat.[57] But the numbers
standing for temperature, thus, are made proportional to the tensions of
gases, that is, to the work done by gases, with otherwise equal changes
of volume. It thus happens that _temperature heights_ and _level heights
of work_ are proportional to one another.
If properties of the thermal condition varying greatly from the tensions
of gases had been chosen, this relation would have assumed very
complicated forms, and the agreement between heat and the other energies
above considered would not subsist. It is very instructive to reflect
upon this point. A _natural law_, therefore, is not implied in the
conformity of the behavior of the energies, but this conformity is
rather conditioned by the uniformity of our modes of conception and is
also partly a matter of good fortune.
VI. THE DIFFERENCES OF THE ENERGIES AND THE LIMITS OF THE PRINCIPLE OF
ENERGY.
Of every quantity of heat _Q_ which does work in a reversible process
(one unaccompanied by loss) between the absolute temperatures _T₁_,
_T₂_, only the portion
_(T₁-T₂)/T₁_
is transformed into work, while the remainder is transferred to the
lower temperature-level _T₂_. This transferred portion can, upon the
reversal of the process, with the same expenditure of work, again be
brought back to the level _T₁_. But if the process is not reversible,
then more heat than in the foregoing case flows to the lower level, and
the surplus can no longer be brought back to the higher level _T₂_
without some _special_ expenditure. W. Thomson (1852), accordingly, drew
attention to the fact, that in all non-reversible, that is, in all real
thermal processes, quantities of heat are lost for mechanical work, and
that accordingly a dissipation or waste of mechanical energy is taking
place. In all cases, heat is only partially transformed into work, but
frequently work is wholly transformed into heat. Hence, a tendency
exists towards a diminution of the _mechanical_ energy and towards an
increase of the _thermal_ energy of the world.
For a simple, closed cyclical process, accompanied by no loss, in which
the quantity of heat _Q₁_ is taken from the level _T₁_, and the quantity
_Q₂_ is deposited upon the level _T₂_, the following relation, agreeably
to equation (2), exists,
_-(Q₁/T₁) + (Q₂/T₂) = 0_.
Similarly, for any number of compound reversible cycles Clausius finds
the algebraical sum
_[sum]Q/T = 0_,
and supposing the temperature to change continuously,
_[integral]dQ/T = 0_ (4)
Here the elements of the quantities of heat deducted from a given level
are reckoned negative, and the elements imparted to it, positive. If the
process is not reversible, then expression (4), which Clausius calls
_entropy_, increases. In actual practice this is always the case, and
Clausius finds himself led to the statement:
1. That the energy of the world remains constant.
2. That the entropy of the world tends toward a maximum.
Once we have noted the above-indicated conformity in the behavior of
different energies, the _peculiarity_ of thermal energy here mentioned
must strike us. Whence is this peculiarity derived, for, generally every
energy passes only partly into another form, which is also true of
thermal energy? The explanation will be found in the following.
Every transformation of a special kind of energy _A_ is accompanied with
a fall of potential of that particular kind of energy, including heat.
But whilst for the other kinds of energy a transformation and therefore
a loss of energy on the part of the kind sinking in potential is
connected with the fall of the potential, with heat the case is
different. Heat can suffer a fall of potential without sustaining a loss
of energy, at least according to the customary mode of estimation. If a
weight sinks, it must create perforce kinetic energy, or heat, or some
other form of energy. Also, an electrical charge cannot suffer a fall of
potential without loss of energy, i. e., without transformation. But
heat can pass with a fall of temperature to a body of greater capacity
and the same thermal energy still be preserved, so long as we regard
_every quantity_ of heat as energy. This it is that gives to heat,
besides its property of energy, in many cases the character of a
material _substance_, or quantity.
If we look at the matter in an unprejudiced light, we must ask if there
is any scientific sense or purpose in still considering as energy a
quantity of heat that can no longer be transformed into mechanical work,
(for example, the heat of a closed equably warmed material system). The
principle of energy certainly plays in this case a wholly superfluous
rôle, which is assigned to it only from habit.[58] To maintain the
principle of energy in the face of a knowledge of the dissipation or
waste of mechanical energy, in the face of the increase of entropy is
equivalent almost to the liberty which Black took when he regarded the
heat of liquefaction as still present but latent.[59] It is to be
remarked further, that the expressions "energy of the world" and
"entropy of the world" are slightly permeated with scholasticism. Energy
and entropy are _metrical_ notions. What meaning can there be in
applying these notions to a case in which they are not applicable, in
which their values are not determinable?
If we could really determine the entropy of the world it would represent
a true, absolute measure of time. In this way is best seen the utter
tautology of a statement that the entropy of the world increases with
the time. Time, and the fact that certain changes take place only in a
definite sense, are one and the same thing.
VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.
We are now prepared to answer the question, What are the sources of the
principle of energy? All knowledge of nature is derived in the last
instance from experience. In this sense they are right who look upon the
principle of energy as a result of experience.
Experience teaches that the sense-elements [alpha beta gamma delta ...]
into which the world may be decomposed, are subject to change. It tells
us further, that certain of these elements are _connected_ with other
elements, so that they appear and disappear together; or, that the
appearance of the elements of one class is connected with the
disappearance of the elements of the other class. We will avoid here the
notions of cause and effect because of their obscurity and
equivocalness. The result of experience may be expressed as follows:
_The sensuous elements of the world ([alpha beta gamma delta ...]) show
themselves to be interdependent._ This interdependence is best
represented by some such conception as is in geometry that of the mutual
dependence of the sides and angles of a triangle, only much more varied
and complex.
As an example, we may take a mass of gas enclosed in a cylinder and
possessed of a definite volume ([alpha]), which we change by a pressure
([beta]) on the piston, at the same time feeling the cylinder with our
hand and receiving a sensation of heat ([gamma]). Increase of pressure
diminishes the volume and increases the sensation of heat.
The various facts of experience are not in all respects alike. Their
common sensuous elements are placed in relief by a process of
abstraction and thus impressed upon the memory. In this way the
expression is obtained of the features of _agreement_ of extensive
groups of facts. The simplest sentence which we can utter is, by the
very nature of language, an abstraction of this kind. But account must
also be taken of the _differences_ of related facts. Facts may be so
nearly related as to contain the same kind of a [alpha beta gamma ...],
but the relation be such that the [alpha beta gamma ...] of the one
differ from the [alpha beta gamma ...] of the other only by the number
of equal parts into which they can be divided. Such being the case, if
rules can be given for deducing _from one another_ the numbers which are
the measures of these [alpha beta gamma ...], then we possess in such
rules the _most general_ expression of a group of facts, as also that
expression which corresponds to all its differences. This is the goal of
quantitative investigation.
If this goal be reached what we have found is that between the [alpha
beta gamma ...] of a group of facts, or better, between the numbers
which are their measures, a number of equations exists. The simple fact
of change brings it about that the number of these equations must be
smaller than the number of the [alpha beta gamma ...]. If the former be
smaller by one than the latter, then one portion of the [alpha beta
gamma ...] is _uniquely_ determined by the other portion.
The quest of relations of this last kind is the most important function
of special experimental research, because we are enabled by it to
complete in thought facts that are only partly given. It is self-evident
that only experience can ascertain that between the [alpha beta gamma
...] relations exist and of what kind they are. Further, only experience
can tell that the relations that exist between the [alpha beta gamma
...] are such that changes of them can be reversed. If this were not the
fact all occasion for the enunciation of the principle of energy, as is
easily seen, would be wanting. In experience, therefore, is buried the
ultimate well-spring of all knowledge of nature, and consequently, in
this sense, also the ultimate source of the principle of energy.
But this does not exclude the fact that the principle of energy has also
a logical root, as will now be shown. Let us assume on the basis of
experience that one group of sensuous elements [alpha beta gamma ...]
determines _uniquely_ another group [lambda mu nu ...]. Experience
further teaches that changes of [alpha beta gamma ...] can be
_reversed_. It is then a logical consequence of this observation, that
every time that [alpha beta gamma ...] assume the same values this is
also the case with [lambda mu nu ...]. Or, that purely _periodical_
changes of [alpha beta gamma ...] can produce no _permanent_ changes of
[lambda mu nu ...]. If the group [lambda mu nu ...] is a mechanical
group, then a perpetual motion is excluded.
It will be said that this is a vicious circle, which we will grant. But
psychologically, the situation is essentially different, whether I think
simply of the unique determination and reversibility of events, or
whether I exclude a perpetual motion. The attention takes in the two
cases different directions and diffuses light over different sides of
the question, which logically of course are necessarily connected.
Surely that firm, logical setting of the thoughts noticeable in the
great inquirers, Stevinus, Galileo, and the rest, which, consciously or
instinctively, was supported by a fine feeling for the slightest
contradictions, has no other purpose than to limit the bounds of thought
and so exempt it from the possibility of error. In this, therefore, the
logical root of the principle of excluded perpetual motion is given,
namely, in that universal conviction which existed even before the
development of mechanics and co-operated in that development.
It is perfectly natural that the principle of excluded perpetual motion
should have been first developed in the simple domain of pure mechanics.
Towards the transference of that principle into the domain of general
physics the idea contributed much that all physical phenomena are
mechanical phenomena. But the foregoing discussion shows how little
essential this notion is. The issue really involved is the recognition
of a general interconnexion of nature. This once established, we see
with Carnot that it is indifferent whether the mechanical laws are
broken directly or circuitously.
The principle of the excluded perpetual motion is very closely related
to the modern principle of energy, but it is not identical with it, for
the latter is to be deduced from the former only by means of a definite
_formal conception_. As may be seen from the preceding exposition, the
perpetual motion can be excluded without our employing or possessing the
notion of _work_. The modern principle of energy results primarily from
a _substantial_ conception of work and of every change of physical
condition which by being reversed produces work. The strong need of such
a conception, which is by no means necessary, but in a formal sense is
very convenient and lucid, is exhibited in the case of J. R. Mayer and
Joule. It was before remarked that this conception was suggested to both
inquirers by the observation that both the production of heat and the
production of mechanical work were connected with an expenditure of
substance. Mayer says: "Ex nihilo nil fit," and in another place, "The
creation or destruction of a force (work) lies without the province of
human activity." In Joule we find this passage: "It is manifestly
_absurd_ to suppose that the powers with which God has endowed matter
can be destroyed."
Some writers have observed in such statements the attempt at a
_metaphysical_ establishment of the doctrine of energy. But we see in
them simply the formal need of a simple, clear, and living grasp of the
facts, which receives its development in practical and technical life,
and which we carry over, as best we can, into the province of science.
As a fact, Mayer writes to Griesinger: "If, finally, you ask me how I
became involved in the whole affair, my answer is simply this: Engaged
during a sea voyage almost exclusively with the study of physiology, I
discovered the new theory for the sufficient reason that I _vividly felt
the need of it_."
The substantial conception of work (energy) is by no means a necessary
one. And it is far from true that the problem is solved with the
recognition of the need of such a conception. Rather let us see how
Mayer gradually endeavored to satisfy that need. He first regards
quantity of motion, or momentum, _mv_, as the equivalent of work, and
did not light, until later, on the notion of living force (_mv²/2_). In
the province of electricity he was unable to assign the expression which
is the equivalent of work. This was done later by Helmholtz. The formal
need, therefore, is _first_ present, and our conception of nature is
subsequently gradually _adapted_ to it.
The laying bare of the experimental, logical, and formal root of the
present principle of energy will perhaps contribute much to the removal
of the mysticism which still clings to this principle. With respect to
our formal need of a very simple, palpable, substantial conception of
the processes in our environment, it remains an open question how far
nature corresponds to that need, or how far we can satisfy it. In one
phase of the preceding discussions it would seem as if the substantial
notion of the principle of energy, like Black's material conception of
heat, has its natural limits in facts, beyond which it can only be
artificially adhered to.
FOOTNOTES:
[Footnote 39: Published in Vol. 5, No. I, of _The Monist_, October,
1894, being in part a re-elaboration of the treatise _Ueber die
Erhaltung der Arbeit_, Prague, 1872.]
[Footnote 40: _On Matter, Living Force, and Heat_, Joule:
_Scientific Papers_, London, 1884, I, p. 265.]
[Footnote 41: "Atqui hoc si sit, globorum series sive corona eundem
situm cum priore habebit, eademque de causa octo globi sinistri
ponderosiores erunt sex dextris, ideoque rursus octo illi
descendent, sex illi ascendent, istique globi ex sese _continuum et
aeternum motum efficient, quod est falsum_."]
[Footnote 42: "A igitur, (si ullo modo per naturam fieri possit)
locum sibi tributum non servato, ac delabatur in _D_; quibus positis
aqua quae ipsi _A_ succedit eandem ob causam deffluet in _D_, eademque
ab alia istinc expelletur, atque adeo aqua haec (cum ubique eadem
ratio sit) _motum instituet perpetuum, quod absurdum fuerit_."]
[Footnote 43: "Accipio, gradus velocitatis ejusdem mobilis super
diversas planorum inclinationes acquisitos tunc esse aequales, cum
eorundum planorum elevationes aequales sint."]
[Footnote 44: "Voi molto probabilmente discorrete, ma oltre al veri
simile voglio con una esperienza crescer tanto la probabilità, che
poco gli manchi all'agguagliarsi ad una ben necessaria
dimostrazione. Figuratevi questo foglio essere una parete eretta
all'orizzonte, e da un chiodo fitto in essa pendere una palla di
piombo d'un'oncia, o due, sospesa dal sottil filo _AB_ lungo due, o
tre braccia perpendicolare all'orizzonte, e nella parete segnate una
linea orizontale _DC_ segante a squadra il perpendicolo _AB_, il
quale sia lontano dalla parete due dita in circa, trasferendo poi il
filo _AB_ colla palla in _AC_, lasciata essa palla in libertà, la
quale primieramente vedrete scendere descrivendo l'arco _CBD_, e di
tanto trapassare il termine _B_, che scorrendo per l'arco _BD_
sormonterà fino quasi alla segnata parallela _CD_, restando di per
vernirvi per piccolissimo intervallo, toltogli il precisamente
arrivarvi dall'impedimento dell'aria, e del filo. Dal che possiamo
veracemente concludere, che l'impeto acquistato nel punto _B_ dalla
palla nello scendere per l'arco _CB_, fu tanto, che bastò a
risospingersi per un simile arco _BD_ alla medesima altezza; fatta,
e più volte reiterata cotale esperienza, voglio, che fiechiamo nella
parete rasente al perpendicolo _AB_ un chiodo come in _E_, ovvero in
_F_, che sporga in fuori cinque, o sei dita, e questo acciocchè il
filo _AC_ tornando come prima a riportar la palla _C_ per l'arco
_CB_, giunta che ella sia in _B_, inoppando il filo nel chiodo _E_,
sia costretta a camminare per la circonferenza _BG_ descritta in
torno al centro _E_, dal che vedremo quello, che potrà far quel
medesimo impeto, che dianzi concepizo nel medesimo termine _B_,
sospinse l'istesso mobile per l'arco _ED_ all'altezza
dell'orizzonale _CD_. Ora, Signori, voi vedrete con gusto condursi
la palla all'orizzontale nel punto _G_, e l'istesso accadere,
l'intoppo si metesse più basso, come in _F_, dove la palla
descriverebbe l'arco _BJ_, terminando sempre la sua salita
precisamente nella linea _CD_, e quando l'intoppe del chiodo fusse
tanto basso, che l'avanzo del filo sotto di lui non arivasse
all'altezza di _CD_ (il che accaderebbe, quando fusse più vicino al
punto _B_, che al segamento dell' _AB_ coll'orizzontale _CD_),
allora il filo cavalcherebbe il chiodo, e segli avolgerebbe intorno.
Questa esperienza non lascia luogo di dubitare della verità del
supposto: imperocchè essendo li due archi _CB_, _DB_ equali e
similmento posti, l'acquisto di momento fatto per la scesa nell'arco
_CB_, è il medesimo, che il fatto per la scesa dell'arco _DB_; ma il
momento acquistato in _B_ per l'arco _CB_ è potente a risospingere
in su il medesimo mobile per l'arco _BD_; adunque anco il momento
acquistato nella scesa _DB_ è eguale a quello, che sospigne
l'istesso mobile pel medesimo arco da _B_ in _D_, sicche
universal-mente ogni memento acquistato per la scesa d'un arco è
eguale a quello, che può far risalire l'istesso mobile pel medesimo
arco: ma i momenti tutti che fanno resalire per tutti gli archi
_BD_, _BG_, _BJ_ sono eguali, poichè son fatti dal istesso medesimo
momento acquistato per la scesa _CB_, come mostra l'esperienza:
adunque tutti i momenti, che si acquistano per le scese negli archi
_DB_, _GB_, _JB_ sono eguali."]
[Footnote 45: "Constat jam, quod mobile ex quiete in _A_ descendens
per _AB_, gradus acquirit velocitatis juxta temporis ipsius
incrementum: gradum vero in _B_ esse maximum acquisitorum, et suapte
natura immutabiliter impressum, sublatis scilicet causis
accelerationis novae, aut retardationis: accelerationis inquam, si
adhuc super extenso plano ulterius progrederetur; retardationis
vero, dum super planum acclive _BC_ fit reflexio: in horizontali
autem _GH_ aequabilis motus juxta gradum velocitatis ex _A_ in _B_
acquisitae in infinitum extenderetur."]
[Footnote 46: "Si gravitas non esset, neque aër motui corporum
officeret, unumquodque eorum, acceptum semel motum continuaturum
velocitate aequabili, secundum lineam rectam."]
[Footnote 47: "Si pondera quotlibet, vi gravitatis suae, moveri
incipiant; non posse centrum gravitatis ex ipsis compositae altius,
quam ubi incipiente motu reperiebatur, ascendere.
"Ipsa vero hypothesis nostra quominus scrupulum moveat, nihil aliud
sibi velle ostendemus, quam, quod nemo unquam negavit, gravia nempe
sursum non ferri.--Et sane, si hac eadem uti scirent novorum operum
machinatores, qui motum perpetuum irrito conatu moliuntur, facile
suos ipsi errores deprehenderent, intelligerentque rem eam mechanica
ratione haud quaquam possibilem esse."]
[Footnote 48: "Si pendulum e pluribus ponderibus compositum, atque e
quiete dimissum, partem quamcunque oscillationis integrae
confecerit, atque inde porro intelligantur pondera ejus singula,
relicto communi vinculo, celeritates acquisitas sursum convertere,
ac quousque possunt ascendere; hoc facto centrum gravitatis ex
omnibus compositae, ad eandem altitudinem reversum erit, quam ante
inceptam oscillationem obtinebat."]
[Footnote 49: "Notato autem hic illud staticum axioma etiam locum
habere:
"Ut spatium agentis ad spatium patientis
Sic potentia patientis ad potentiam agentis."]
[Footnote 50: "Cependant, comme dans cet ouvrage on ne fut d'abord
attentif qu'à considérer ce beau développement de la mécanique qui
semblait sortir tout entière d'une seule et même formule, on crut
naturellement que la science etait faite, et qu'il ne restait plus
qu'à chercher la démonstration du principe des vitesses virtuelles.
Mais cette recherche ramena toutes les difficultés qu'on avait
franchies par le principe même. Cette loi si générale, où se mêlent
des idées vagues et étrangères de mouvements infinement petits et de
perturbation d'équilibre, ne fit en quelque sorte que s'obsurcir à
l'examen; et le livre de Lagrange n'offrant plus alors rien de clair
que la marche des calculs, on vit bien que les nuages n'avaient paru
levé sur le cours de la mécanique que parcequ'ils étaient, pour
ainsi dire, rassemblés à l'origine même do cette science.