THE PRESENT PHYSICAL UNIVERSE.
... οἱ οὐρανοὶ ῥοιζηδὸν παρελεύσονται, στοιχεῖα δὲ καυσούμενα
λυθήσονται, καὶ γῆ καὶ τὰ ἐν αὐτῇ ἔργα κατακαήσεται.—Πετρού Βʹ.
γʹ.
‘The cloud-capp’d towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve;
And, like this insubstantial pageant, faded,
Leave not a rack behind.’—SHAKESPEARE, _Tempest_.
‘All worldly shapes shall melt in gloom,
The sun himself must die
Before this mortal shall assume
His immortality.’—CAMPBELL.
92. Having in the last chapter briefly indicated the nature of the
proposition which we intend to bring forward, we must next study, as
a preliminary to further discussion, what science tells us about the
present physical universe: what are the general laws to which it is
now subject; when and what must have been its beginning; when and
what will be its inevitable end.
We have been driven into becoming accustomed to the phrase, ‘the
material universe,’ which is generally used in a sense absolutely
identical with that which we have chosen as the title of this
chapter. We shall soon see that the term is a very inapt one,
inasmuch as matter is (though it may sound paradoxical to say so) the
less important half of the material of the physical universe.
In the present chapter we shall still further restrict ourselves by
omitting, as far as possible, any reference to life (even in its
lowest aspect), and we likewise defer to a future chapter our account
of the more reasonable speculations which have been advanced with
regard to the intimate structure of matter and ether.
93. It is only within the last thirty or forty years that there has
gradually dawned upon the minds of scientific men the conviction that
there is something besides matter or stuff in the physical universe,
something which has at least as much claim as matter to recognition
as an objective reality, though, of course, far less directly
obvious to our senses as such, and therefore much later in being
detected. So long as men spoke of light, heat, electricity, etc., as
imponderables, they merely avoided or put aside the difficulty.
When they attempted to rank them as matter,—heat, for instance, as
caloric,—they at once fell into errors, from which a closer scrutiny
of experimental results would assuredly have saved them. The idea
of _substance_ or _stuff_ as necessary to objective existence very
naturally arises from ordinary observations on matter; and as there
could be little doubt of the physical reality of heat, light, etc.,
these were in early times at once set down as matter. Fire, in fact
(including, it is to be presumed, everything which involved either
heat or flame, real or apparent), was in early times one of the four
so-called _elements_.
In those days the sun was supposed to be only a great fire; a
lightning-flash, an aurora, or a comet, was merely a flame; in other
words, the essence of all these was the element fire, or, as it
was later called, caloric. The sun, except when he appeared as the
spreader of pestilence, was the beneficent fire, as were also some of
the planets; the lightning, the comet, even the moon and Saturn, were
baleful fires.
This endeavour to assign a substantive existence to every phenomenon
is, of course, perfectly natural; but on that very account
excessively likely to be wrong.
_Humanum est errare_ comes with quite as much heart-felt conviction
of its truth from the lips of the honest Pagan as from those of the
Christian believer; though perhaps its meaning may be considerably
less extended in the former than in the latter case.
94. But, before discussing what is that _something else_ besides
stuff which has an _objective_ though not a _substantive_[35]
existence, let us in the first place inquire into the grounds of our
belief, that matter itself has a real existence external to us; that,
in fact, the so-called evidence of our senses is not a mere delusion.
There is a strong temptation to be metaphysical here, but we will
endeavour to resist it.
Now physical science furnishes us with the following among many other
arguments in proof of the reality of the external universe:—
_Experience of the most varied kind consistently shows us that we
cannot produce or destroy even the smallest quantity of matter._
Exercise our greatest powers of imagination, do with it what
we please, we cannot make our senses indicate to us an increase
or diminution in a given quantity of what we call matter. We
find it so far amenable to our control that we can alter its
arrangement, form, density, state of aggregation, temperature,
etc.; nay, by so approximating it to other matter as to produce a
chemical combination, we may entirely transform its appearance and
properties,—_all but one_: its mass or quantity is completely beyond
our control. Measure it by what process we please, by the ‘muscular
sense,’ by weight, anyhow, there it is, altogether independent of us,
laughing our efforts to scorn! Can this be a mere mental idea which
the mind that conceived it (or, at all events, in some way received
the conception of it) is unable to destroy?
But there is one other argument on this point which must be
mentioned. Not only do our own senses invariably indicate to us the
impossibility of altering the quantity of matter, but the senses of
all men alike point to the same quantity, quality, and collocation
of matter in the earth and external to the earth. Whence this
extraordinary agreement between the evidences of the senses in
different men, when the minds are so different?
Our conviction then of the objective reality of matter (at least
from the point of view of the Natural Philosopher) is based upon the
experimental truth that we can neither increase nor diminish its
quantity, in fact on what we may conveniently for our present purpose
call the _Conservation of Matter_.
95. Here let us pause for a moment to compare together this view of
matter and the definition of the laws of the universe, which we have
already given. The laws of the universe we defined (Art. 54) to be
the laws according to which the beings in the universe are trammelled
by the Governor thereof as regards time, space, and sensation. Now,
it may be asked, is this definition consistent with a belief in the
objective reality of matter? Our reply is, that to our minds the two
are in perfect accordance.
We do not here intend to enter into any metaphysical discussion. It
is enough for us to say that our practical working certainty of the
reality of matter depends upon the facts, _firstly_, that it offers
resistance to our imagination and our will, and, _secondly_, that in
particular it offers _absolute_ resistance to all attempts to change
its quantity. We shall soon see that experiment teaches us that both
properties belong to something else.
96. Returning from this digression let us therefore assume that the
objective reality of the external universe has been proved, and
that this reality is strongly impressed upon us in virtue of that
principle which we have called the conservation of matter.
But as soon as we grant this, we are obliged by our reason, however
little our senses may incline us to it, or rather however much they
may dispose us against it, to allow objective reality to whatever
else may be found to be _in the same sense_ conserved. (We have
here italicised these four words for a reason which will afterwards
appear.) This is a question which deserves and must secure careful
consideration.
97. In abstract dynamics several things are said and mathematically
proved by deductions from experiment to be conserved, but one
only of these in the strict sense in which we have spoken of the
conservation of matter. We will examine them briefly, and our
non-mathematical readers must pardon us if we make use of certain
technical expressions belonging to the domain of mathematical physics.
[It is absolutely essential that the reader should have clear notions
on these points, for there is widespread confusion and error as to
the meaning even of so simple and elementary a term as ‘_force_.’
He will often find it used indifferently in either of two senses
which have no connection whatever with one another; and unless he
completely gets over this abuse of language he need not hope to be
able to follow the present portion of our preliminary argument. Force
proper is a pull, push, weight, pressure, etc., and can be measured,
in the vernacular of engineers, as equivalent to so many pounds
weight; but the unjustifiable use of the word applies it to _work
done by a force_, so many pounds raised so many feet, _i.e. force
overcome through a space_. Two such things are of different kinds,
and cannot possibly be compared together. They differ in fact in
precisely the same way as length or breadth differs from superficial
area, _i.e._ as a linear foot differs from a square foot! And the
modern abuse of the word is more outrageous, alike to science and
to common sense, than would be the attempt to assign the height of
a mountain in acres! For the absurdity does not end even here. We
have, as yet, absolutely no proof whatever that force proper has
objective existence. In all probability there is no such _thing_ as
force (which is suggested to us by the impressions of our _muscular_
sense), any more than there is such a _thing_ as Sound, or Light,
which are mere names for physical impressions produced upon special
nerves by the energy of undulatory motions of certain media. The
term, however, is a very convenient one for the rate of transference
or transformation of energy per unit of length in a given direction.]
(1.) _Conservation of Momentum._—What is understood by this is a
mere direct consequence of Newton’s _first_ interpretation of his
Third Law of Motion, viz., that _Action and Reaction are equal and
opposite_. In this _first_ interpretation Newton tells us to consider
actions and reactions as forces proper, or (their equivalents)
quantities of motion. This is the term employed by Newton; but we
now designate it _momentum_, and measure it by the product of the
mass and the velocity of a body. Stated in its simplest form, this
law asserts that the momentum of a system of bodies, measured in any
direction whatever, is not altered by their mutual action, whether
that action be of the nature of traction, attraction, repulsion, or
impact. And we see at once from this third law of motion that it must
be so, because the change of momentum, in any direction, of any one
part of the system, per unit of time, is the measure of the force
acting on that part in that direction. Whatever momentum in this
particular direction is gained by one member of the system must have
been lost by other members, but not from their whole momentum, merely
from the part of it in this direction. It thus appears that the
(algebraic) sum of the momenta generated by the mutual actions of the
system is zero.
These momenta are in fact _directed magnitudes_ (like the forces of
which they are the measure), and are therefore capable of _cancelling
one another_ when their numerical amounts are equal and their
directions are opposite. In this sense the conservation is of the
same nature as that of the imagined electric or magnetic fluids,
where no portion whatever of one kind can be produced without the
simultaneous appearance of an equal quantity of the other, a quantity
just capable of neutralising it. This is obviously not in any sense
analogous to the Conservation of Matter of which we have just spoken.
As an illustration take a loaded cannon. Before firing, neither
cannon nor ball had momentum. After firing, the ball has a certain
momentum, the cannon (in virtue of its recoil) an _equal_ and
opposite momentum. If we could exactly reverse the motions of the
cannon and ball just as they separate, the impact between them would
just reduce each to rest, and no momentum would be left. Considered
separately after the discharge, each has momentum, but _in the
complete system_ of cannon and ball there is no momentum—there being
equal quantities of positive and negative, in the same line. In fact
momentum cannot be produced or destroyed _in any system as a whole_.
This is the Conservation referred to. It is as if a man always when
he received a sum of money fell to the same amount in debt—the state
of his affairs, as shown by his books, would of course not be altered.
(2.) _Conservation of Moment of Momentum_.—Here we deal with
quantities of the order of the moments of forces about an axis, _i.e.
couples_ in Poinsot’s sense. These also are directed magnitudes
depending for their conservation upon the first interpretation of
Newton’s third law, and therefore the same remarks apply to them as
to the preceding.
(3.) _Conservation of Vis Viva_.—_Vis viva_ is the old name for
energy of motion or the consequent power of doing work. We now deal
with quantities which cannot possess direction, because they are
essentially _products of pairs of quantities similarly directed_,
and are therefore all to be treated as of the same algebraic sign,
or rather (to adopt the language of Sir W. R. Hamilton) as signless
quantities. With such there can of course be no cancelling.
To make our meaning clear, let us consider upon what _vis viva_
depends. It depends upon and is proportional to the product of
the mass into the _square_ of the velocity. Compare, or rather
contrast, this with the definition of momentum given above, and it
will be seen that _vis viva_ is the product of the momentum and the
velocity. Now mass is of course a signless quantity; evidently we
cannot have negative mass. Then with regard to the square of the
velocity, this will be positive whether the velocity be positive
or negative, whether it be in one direction or the opposite. _Vis
viva_, therefore, or energy of motion, is something which is not
affected with the sign of direction, or, as we have already said,
it is a signless quantity. It is found to be convenient to measure
it as _half_ the product of the moving mass into the square of its
velocity. So measured, it is now called (see § 99) _kinetic energy_.
Now to our cannon again. Before firing there is no _vis viva_ of
either cannon or ball. After firing each has _vis viva_, but that
of the ball is greater than that of the cannon in the proportion in
which the cannon’s mass exceeds that of the ball. And the system as
a whole has _vis viva_ though it has no momentum. If, as before, we
could reverse the motions of cannon and ball, then, even when they
impinged, the _vis viva_ would _not_ be lost. As will presently be
seen, it would be employed in heating both the impinging bodies.
98. We have said that the energy which a body contains—its _vis
viva_—its power of doing work, is independent of the direction in
which it is moving; and, further, that while the mass is the same, it
is proportional to the square of the velocity. For instance, we may
measure the energy of a cannon-ball or of an arrow by the distance
it will carry itself up against the force of gravity, represented by
its own weight, when shot vertically upwards, and we find that with
a double velocity it will go four times as high. Or we may point the
cannon horizontally, and measure the energy of the same ball by the
number of planks of oak wood which it can penetrate, and we shall
find that a ball with double the velocity will penetrate nearly four
times as many as one with the single velocity. All such experiments
concur together in convincing us that the energy of the ball is
independent of the direction in which the cannon is pointed, and is
proportional to the square of the velocity, so that a double velocity
will give a fourfold energy.
99. We have just now spoken about a cannon-ball fired into the
air against the force of gravity. Such a ball, as it mounts, will
each moment lose part of its velocity, until it finally comes to
a standstill, after which it will begin to descend. When it is
just turning it is perfectly harmless, and if we were standing on
the top of a cliff to which it had just reached, we might without
danger catch it in our arms and lodge it on the cliff. Its energy
has apparently disappeared. Let us, however, see whether this is
really true or not. It was fired up at us, let us say, by a foe at
the bottom of the cliff, and the thought occurs to us to drop it down
upon him again, which we do with great success, for he is smashed to
pieces by the ball.
In truth, dynamics informs us that such a ball will again strike the
ground with a velocity, and therefore with an energy precisely equal
to that with which it was originally projected upwards. Now, when at
the top of the cliff, if it had not the energy due to actual motion,
it had nevertheless some sort of energy due to its elevated position,
for it had obviously the power of doing work. A pond of still water,
unless it can fall, _i.e._ unless it has what is technically called
a ‘_head_,’ is of no use in driving a water-wheel. The head, or
the power of descending, gives it a store of dormant energy, which
becomes active as the water gradually descends. And the same amount
of work may be obtained (by means of a turbine for instance) from
a small quantity of water, provided it has a great ‘head,’ as can
be obtained (by means of an ordinary overshot or breast wheel) from
water with far less head, provided it be supplied in proportionally
greater quantity. _We thus recognise two forms of energy which change
into one another, the one due to actual motion and the other to
position; the former of these is generally called kinetic, and the
latter potential energy._
All this appears to have been clearly perceived by Newton,
who gave it as a _second_ interpretation of his Third Law of
Motion. His statement is equivalent, in modern language, to
the following:[36]—_Work done on any system of bodies has its
equivalent in the form of work done against friction, molecular
forces, or gravity, if there be no acceleration; but if there be
acceleration, part of the work is expended in overcoming resistance
to acceleration, and the additional kinetic energy developed is
equivalent to the work so spent._
100. Thus Newton expressly tells us (though not in these words) that
we are to include in the same category work done by or against a
force—whether that force be due to gravity, friction, or molecular
action (such as elasticity, for instance), or even to acceleration.
(_a._) When work is done against gravity, as in lifting a mass from
the ground, we have just seen that it is (as it were) stored up
in the raised mass; we can recover it at any time by letting the
mass descend. Thus it is that we furnish a clock with motive power
sufficient to keep it going for a week in spite of friction and other
resistance, by simply winding up its weights.
(_b._) When work is done against molecular forces, we have a similar
storing up, as, for instance, in drawing a bow or in winding up a
watch.
(_c._) When work is done against the inertia of a body, _i.e._
to accelerate its velocity, Newton’s definitions show that the
additional kinetic energy so produced is equal to the work so spent.
(_d._) In abstract dynamics we simply consider as lost the work spent
against friction. In Newton’s time it was not known what became of it.
101. Leaving out, then, for the present, the fourth alternative,
we see that whatever work is spent, we must, according to Newton,
even in abstract dynamics recognise that _it is not lost, but only
transformed_ into an equivalent quantity stored up for future use,
either in a quiescent form (as, for instance, the potential energy of
a raised weight or bent spring), or in an active form (as the kinetic
energy of a moving mass). Here, then, at last, we recognise the
same sort of conservation as that which we found in matter. But the
statement so far is defective, as we have seen, in one particular.
What becomes of work spent in overcoming friction? or what becomes of
the energy of the blacksmith’s hammer after it has struck the anvil?
To this experiment alone can give the answer. Let us see what it has
told us.
Man has been called a reasoning animal, a laughing animal, etc.,
according to the momentary whim or humour of the classifier; but he
is perhaps still more definitely separated from all other animals
when specified as _the_ ‘cooking animal.’ Now, it has always appeared
to us as something little short of marvellous that, even for the high
purpose of cooking his food, or of inflicting exquisite torture on a
vanquished foe, savage man should ever have hit upon the process of
procuring fire by friction. Considering his condition, and comparing
his opportunities and his success with those of even our greatest
modern physicists, we cannot but look upon this as one of the very
greatest and most notable discoveries ever made in physics. All the
more notable, too, from the fact that a man like Newton, though of
course aware of it, absolutely missed its significance even at the
very moment when it alone was wanted to fill a serious _lacuna_ in
one of his grandest and most important practical generalisations.
The missing link was all but supplied by Rumford and Davy at the
very end of last century. Rumford’s boiling of water by the heat
generated in the boring of a cannon, and Davy’s melting of ice by
friction _in vacuo_, were each conclusively demonstrative alike of
the non-materiality of heat and of the ultimate fate of work spent in
friction, which is thus seen to be converted into heat; or at least
these experiments could easily have been made demonstrative by very
slight additions to, or modifications of, their author’s methods or
reasoning. But the exact and formal enuntiation of the equivalence of
heat and work required to fill the _lacuna_ in Newton’s statement was
first given by Davy in 1812.
102. Let us here pause for a moment and contemplate the position to
which the solution of our problem had even then attained. Visible
kinetic energy, such as that of a cannon-ball shot upwards, is
transformed as it rises into visible potential energy. As the ball
descends its energy is retransformed from the potential into the
kinetic variety until, when it is about to strike the earth, it has,
or rather would have if there were no atmosphere, as much kinetic
energy as it had when it was first shot upwards.
When the ball has once struck the earth its kinetic energy of visible
motion is changed by impact into that kinetic energy of invisible
motion of its particles which is called heat; and, generally
speaking, in all cases of friction, percussion, and atmospheric
resistance we have a change of visible energy into heat, as for
instance when a railway train is stopped by the action of the
brake, when a blacksmith strikes the anvil with his hammer, when a
cannon-ball moves through and heats the air, or when a meteorite or
falling-star is rendered incandescent by the resistance it meets with
even in the higher and rarer strata of the atmosphere.
We had thus come to the stage of regarding heat as a species of
molecular energy into which visible energy is often transformed, and
very soon afterwards it came to be perceived that there were other
forms of molecular energy besides heat—some of these being potential
and some kinetic. Thus two substances may possess mutual chemical
affinity when separated from each other, just as a raised stone
tends to fall again to the earth, and we obtain a form of potential
energy in the one case as truly as in the other. When, for instance,
we have carbon or coal in our cellars or our mines, and oxygen in
the air, we are in possession of a store of chemical potential
energy upon which we can draw at any moment and change it during the
process of combustion from the potential to the kinetic form. Again,
in a current of electricity we have no doubt a species of kinetic
energy, although it still puzzles men of science to say what form of
invisible motion such a current implies. From all this, without being
further perplexed with scientific details, our readers will perceive
that there are many different forms, some of them potential, and
others of them kinetic, in which energy may appear.
While we were thus grasping the fact that energy can appear under
various forms, we were also beginning to perceive that it had great
powers of transmutation—going about from one form to another, and Sir
W. R. Grove did good work at this stage of the inquiry in bringing
together the various cases of such transmutations in his work on the
Correlation of the Physical Forces.
In spite of this, it was left for Joule and Colding, who worked
almost simultaneously and by well-devised experimental methods from
about the year 1840, independently to discover, and by degrees to
enuntiate, by means of arguments founded on _the only admissible
basis—experiment_, the grand law of the _Conservation of Energy_. In
its most general form, the statement of the conservation of energy
is merely a completed version of the passage we have already quoted
from Newton; and the experimental discoveries of Rumford and Davy,
extended and completed by Joule and Colding, allow us now to put
Newton’s second or alternative interpretation of his _Third Law of
Motion_ into the modern statement of the _Conservation of Energy_.
_In any system of bodies whatever, to which no energy is communicated
by external bodies, and which parts with no energy to external
bodies, the sum of the various potential and kinetic energies remains
for ever unaltered._
In other words, while the one form of energy becomes changed into
the other,—potential into kinetic and kinetic into potential, or
one species of either into another;—yet each change represents at
once a creation of one kind of energy and a simultaneous and equal
annihilation of another, the total energy present, as we have already
said, remaining for ever unaltered.
103. Taking as our ‘system of bodies’ the whole physical universe,
we now see that, according to the test we have already laid down,
energy has as much claim to be regarded as an objective reality as
matter itself. But the forms of statement are most markedly different
for the two. We before spoke of the quantity of matter without
qualification, but we now speak of _the sum of the two kinds_ of
energy. Let us think for a moment of this, and we see that whereas
(to our present knowledge, at least) matter is always the same,
though it may be masked in various combinations, energy is constantly
changing the form in which it presents itself. The one is like the
eternal, unchangeable Fate or _Necessitas_ of the antients; the other
is Proteus himself in the variety and rapidity of its transformations.
Φύσις, διαδόχαις σχημάτων τρισμυρίοις
ἀλλάσσεται τύπωμα, Πρωτέως δίκην,
πάντων ὅσ’ ἔστι ποικιλώτατον τέρας·
τῆς δ’ αὖτ’ Ἀνάγκης ἐστ’ ἀκίνητον σθένος,
μόνη δ’ ἁπάντων ταὐτὸ διαμένουσ’ ἀεὶ
βροτῶν τε καὶ θεῶν πάντ’ ἀποτρύει γένη.[37]
104. And again, _energy is of use to us solely because it is
constantly being transformed_. When the sluice is shut, or the fire
put out, the machinery stops; when a man cannot digest his food,
he breaks down altogether. Coal in itself, except on account of an
occasional fossil it may contain, or its still somewhat uncertain
mode of formation, or (to take a lower point of view) as a material
for ornament, is a very useless thing indeed: its grand value
consists in its chemical affinity, in virtue of which it possesses
great potential energy as regards the oxygen of the air, which
can very easily be transformed into its equivalent in heat. ‘Keep
your powder dry’ is merely one way of saying ‘preserve the ready
transformability of your energy.’ In fact, if we think for a moment
over what has just been said, to the effect that the only real things
in the physical universe are matter and energy, and that of these
matter is simply passive, it is obvious that all the physical changes
which take place, including those which are inseparably associated
with the thoughts as well as the actions of living beings, are merely
transformations of energy. Thus it is an inquiry of the very utmost
importance as regards the present universe: _Are all forms of energy
equally susceptible of transformation?_ To see the importance of this
question, the reader has only to reflect that if there be any one
form of energy less readily or less completely transformable than the
others, and if transformations constantly go on, more and more of the
whole energy of the universe will inevitably sink into this lower
grade as time advances. Hence the whole possibility of transformation
must steadily grow less and less; in scientific language, though the
_quantity_ of energy remains for ever unchanged, its _availability_
steadily decreases.
105. Now, every one knows a case in which there may be an unlimited
amount of energy present, no part of which is available for
transformation. It is the simple one of heat in a number of bodies,
_when all are at the same temperature_. To obtain work from heat we
must have hotter and colder bodies, to correspond, as it were, with
the boiler and condenser of a heat-engine; and just as we can get no
work from still water if it be all at the same level, _i.e._ if no
part of it can fall, so in like manner we can get no work from heat
unless part of it can fall from a higher to a lower temperature.
This is a remark of the very utmost consequence to our argument,
and must therefore be fully elucidated. Unfortunately it is not as
yet possible to do this without introducing a good many scientific
technicalities which are unsuited to the great majority of readers.
In the next eight sections we endeavour to explain it as simply as we
can. The reader who cannot easily follow us may pass, without break
of continuity in the argument, at once to Art. 114.
* 106. The first step in the investigation of the transformation of
heat into work was taken by Sadi Carnot in 1824: a step which has
recently been found of inestimable value in every branch of modern
physical science. He devised a method of startling originality for
the purpose of attacking this special question of the production of
work from heat. His inferences from its application were not all
correct; this was due however to no fault of the method, but to the
fact that he unfortunately assumed (though with caution, and under
a protest almost amounting to an assertion of the opposite) the
materiality of heat. His method embraces two perfectly new ideas:—
(1.) That, at least with our present knowledge, no inference is
possible as to the relation between heat and work, until the heated
or working substance is brought back, after a complete _Cycle_ of
operations, to its initial physical state.
Obvious as this statement, once made, is, it was altogether ignored
(twenty years after Carnot) by Séguin and Mayer, whom some authors
still persist in setting forth as the founders of the dynamical
theory of heat. Their speculations were entirely vitiated by their
violation of this principle.
(2.) That an engine whose cycle of operations is reversible is a
perfect engine, that is to say, gives the greatest possible amount of
work from a given quantity of heat with any assigned temperatures of
boiler and condenser.
The term reversible is not here used in the popular sense in
which a mere reversal of the direction of motion of each part is
contemplated, _i.e._ what would be more properly termed ‘backing,’
it is used in the higher sense of taking an engine which converts a
certain quantity of the heat spent on it into work, while it lets the
rest down from the boiler to the condenser, and then spending upon
it the same amount of work with the result of taking back the heat
from the condenser, adding thereto the heat-equivalent of the work so
spent, and thus restoring the whole of its original loss in heat to
the boiler; simply in fact reversing all the results of the direct
action.[38]
* 107. Sir W. Thomson, in 1848, was the first to recall attention
to the work of Carnot, after Colding and Joule had published their
experimental discoveries; and he pointed out that the action of
the reversible engine gave what had been up to that time vainly
sought, an absolute definition of temperature—a definition, that is,
altogether independent of the properties of any particular species of
matter. In fact it is obvious that as reversibility in the sense we
have just explained is the stamp of perfection in a heat-engine, all
reversible engines, whatever be the working substance, will, under
the same circumstances, that is to say, _with the same temperatures
of boiler and condenser_, convert the same fraction of the heat spent
on them into work. This, of course, still leaves wide scope for
choice of a definition of temperature: but that finally determined
on by Thomson was chosen (in consequence of a hint from some
experimental results of Joule) so as to make the absolute measurement
agree nearly with that of the long-familiar air-thermometer. It
therefore stands as follows:—
_The heat taken in by a perfect engine is to the heat given out by it
in the same proportion as the absolute temperature of the boiler to
that of the condenser._
Of course it is hardly necessary to state that it is only the excess
of the heat taken in over that given out by any engine that can have
been converted into available work. This follows at once from the
conservation of energy.
Experiments carried on by Joule and Thomson[39] together have shown
that the absolute zero of temperature is nearly 274° below zero of
the centigrade scale; so that on the absolute scale the temperature
of melting ice is 274°, while that of water boiling under the
standard pressure is 374°.
* 108. In 1849 James Thomson made a very remarkable application of
Carnot’s reasoning, the first of a series of such applications which
have since done immense service in the extension of almost every
branch of physics. He showed in fact that, because water expands in
the act of freezing, _the melting point of ice must be lowered by
pressure_. Sir W. Thomson in the same year verified this deduction,
to its numerical details, by direct experiment. Trifling as the
predicted and measured effect appears (one degree centigrade for each
2000 lbs. additional pressure per square inch), there can now be no
doubt that it goes at least very far to explain the varied effects of
the extraordinary plasticity of glacier-ice so beautifully made out
by the direct measurements of Forbes.
* 109. We have said that Carnot unfortunately based his reasoning on
the assumed materiality (and therefore indestructibility) of heat. It
therefore became a question of great importance to find how properly
to adapt his methods to the true theory. James Thomson’s verified
prediction had already given a correct and absolutely new physical
result from Carnot’s principles. How then must we get rid of his
false assumption?
Clausius attempted this in 1850, but his method is based solely upon
the observed fact that in general heat tends from hotter to colder
bodies. This we know is not always the case, for a fine wire may
be made red-hot by the current from a thermo-electric battery (of
a sufficient number of pairs) where ice and boiling water alone are
used to cool and heat the alternate junctions. Here heat certainly
passes from colder bodies to a hotter one. Clausius, no doubt,
several years later, extended his original statement, so as to make
it stand thus:—Heat cannot _of itself_ pass from a colder to a hotter
body. We do not consider even this sufficiently obvious for an axiom,
were it certainly true, but, as will be seen presently, it is not. In
fact the so-called axiom is constantly being violated, though on a
very small scale, in every mass of gas.
* 110. It was Sir W. Thomson[40] who (in 1851) first correctly
adapted Carnot’s magnificently original methods to the true theory
of heat; and it is especially noteworthy to remark how, even at that
early time, he saw the full danger of attempting to lay down anything
too definite on the subject. The following is the axiom he gives:—
‘_It is impossible by means of inanimate material agency to derive
mechanical effect from any portion of matter by cooling it below the
temperature of the coldest of the surrounding objects._’
But he appends the following guarded note:—
‘If this axiom be denied for all temperatures, it would have to be
admitted that a self-acting machine might be set to work and produce
mechanical effect by cooling the sea or earth, with no limit but the
total loss of heat from the earth and sea, or, in reality, from the
whole material world.’
The full importance of this will appear presently.
To those who can accept Thomson’s axiom with the explanation appended
to it, Carnot’s proposition that a reversible engine is perfect (in
the sense of being the best possible) is demonstrated at once, as
follows, _ex absurdo_.
Suppose there could be an engine, M, more perfect than a reversible
engine, N. Set the two to work together as a compound engine, M
letting down heat from boiler to condenser, and doing work; N
spending work in pumping back again the heat to the boiler. If N
be made to restore to the boiler at every stroke exactly what M
takes from it, the compound engine will do external work, for, by
hypothesis, M is more perfect than N. Whence does the work come?
Not from the boiler, for it remains as it was. Hence N must take
more heat from the condenser than M gives it; _i.e._ you get work by
cooling the condenser.
Carry the reasoning a little further, and we see that if the excess
of work given by M were spent upon N, and thus no work on the whole
either spent or given out, the condenser would be still further
cooled, and the boiler heated! This, to most people, would seem to
imply an ample _reductio ad absurdum_. But Clerk-Maxwell has shown
it to be physically possible, and has thus thoroughly justified
Thomson’s caution about his axiom. As this is a point of very great
importance, we offer no excuse for treating it pretty fully.
* 111. Clerk-Maxwell’s reasoning is given as depending upon the
molecular theory of gases, but the only necessity for so restricting
it appears to be that we thereby connect the reasoning more directly
with _Heat_, which, on this theory, is supposed to be the energy
of motion of the molecules of the gas. The illustration, however,
is more general, and at the same time more simple, if we do not
at first refer either to heat or to the molecular hypothesis of
the constitution of gases, but treat the question simply as one
concerning the possible motions of a number of little material
particles.
Assume, then, that a great number of small equal spherical particles
of matter are enclosed in a vessel of any form, and assume further
that (either by collision or by repulsive force) each of these
has the power of rebounding from another or from the wall of the
vessel, as if it were elastic, and had unit _co-efficient of
restitution_,[41] as defined in treatises on natural philosophy.
Then it can be shown, as a matter of direct calculation, that—start
these particles as we please, in all sorts of directions, and with
velocities as varied as we please—after a time, which will be
shorter as the number of particles is greater, a sort of permanent
state will be arrived at in which a certain law of distribution of
velocity prevails among the particles (the same law as that of the
_Probability of Error_, as it is technically called), the greater
number of them having nearly the mean square velocity, and those
which have much less or more than that being fewer and fewer as
the defect or excess is greater. The tendency is to an average
distribution of these varieties of velocity throughout the vessel,
and the impacts on the sides will thus be nearly the same on every
square inch of its surface. After this there is—_always provided
the particles be sufficiently numerous_—no perceptible change in the
statistics of the group, except in so far as concerns _individual_
particles, which may sometimes be moving with great, sometimes with
very small, velocity, but which, in the long-run, will far more
often be moving with the mean square velocity, or at least some
velocity very near it. Hence, in no part of the vessel will the
average energy be sensibly greater than in another, and therefore
(so far as the contents of the vessel alone are concerned) there is
no possibility of getting work from them. But by enlisting in our
service conceivable finite beings (imagined by Clerk-Maxwell, and
called demons by Thomson), it would be possible materially to alter
this state of things, even although these beings should do absolutely
no work.
* 112. For suppose a firm partition, full of little doors (themselves
without mass) to be placed so as to divide the vessel into two,
and set a demon at each door, with instructions to open it for an
instant whenever he sees he can thereby let a quick-moving particle
escape from the first compartment to the second, or a slow-moving
particle from the second into the first. Then, _because the tendency
is not to a uniform distribution of velocity_ among the particles,
but to a distribution which involves quicker and slower in certain
proportions, we may imagine this process to be carried on long enough
to make a considerable difference in the average velocities of the
particles in the two compartments, though the numbers of particles in
each compartment may remain almost unchanged. The consequence will
of course be a greater pressure per square inch on the walls of the
second compartment than of the first; and thus, if the partition wall
were moveable, a certain amount of work might be obtained by allowing
it to move. Thus a group of particles originally incapable, without
external assistance, of doing work, may be rendered capable of doing
work by mere _guidance_ applied by finite intelligence.
* 113. Now let us refer for a moment to the molecular theory of
gases, and we see that what the demons (without any expenditure of
work, each being, so far as he is required, virtually a combination
of two intelligent perfect engines, one working direct, the other
reversed) have guided the gas to do, is to transfer heat from a
colder to a hotter portion of the gas.
The only reason why this does not occur without the assistance of
demons (at least to an extent, or for a length of time, sufficient to
produce a sensible effect) lies in the enormous number of particles
per cubic inch in even the most rarefied gas. Hence, _solely because
of the excessive numbers and minuteness of the particles of matter_,
the one chance of escape from Carnot’s proposition is denied us, and
therefore we must allow that, so far as the physical universe is
concerned, a reversible heat-engine is the best possible.
But if a reversible heat-engine be the best possible, then the
principle which we have italicised in Art. 107 must hold good, and
from this it follows that only a portion of the heat passing through
a perfect engine can be transformed into useful work unless the
condenser of the engine be at the absolute zero of temperature—a
condition which can never be attained.
114. It thus appears that at each transformation of heat-energy into
work a large portion is degraded, while only a small portion is
transformed into work. So that while it is very easy to change all
of our mechanical or useful energy into heat, it is only possible
to transform a portion of this heat-energy back again into work.
After each change too the heat becomes more and more dissipated
or degraded, that is, less and less available for any future
transformation.
In other words, the tendency of heat is towards equalisation; heat is
_par excellence_ the communist of our universe, and it will no doubt
ultimately bring the present system to an end. The visible universe
may with perfect truth be compared to a vast heat-engine, and this is
the reason why we have brought such engines so prominently before our
readers. The sun is the furnace or source of high-temperature heat of
our system, just as the stars are for other systems, and the energy
which is essential to our existence is derived from the heat which
the sun radiates, and represents only an excessively minute portion
of that heat. But while the sun thus supplies us with energy he is
himself getting colder, and must ultimately, by radiation into space,
part with the life-sustaining power which he at present possesses.
Besides the inevitable cooling of the sun we must also suppose that
owing to something analogous to ethereal friction[42] the earth
and the other planets of our system will be drawn spirally nearer
and nearer to the sun, and will at length be engulfed in his mass.
In each case there will be, as the result of the collision, the
conversion of visible energy into heat, and a partial and temporary
restoration of the power of the sun. At length, however, this
process will have come to an end, and he will be extinguished until,
after long but not immeasurable ages, by means of the same ethereal
friction his black mass is brought into contact with that of one or
more of his nearer neighbours.
115. Not much further need we dilate on this. It is absolutely
certain that life, so far as it is physical, depends essentially upon
transformations of energy; it is also absolutely certain that age
after age the possibility of such transformations is becoming less
and less; and, so far as we yet know, the final state of the present
universe must be an aggregation (into one mass) of all the matter it
contains, _i.e._ the potential energy gone, and a practically useless
state of kinetic energy, _i.e._ uniform temperature throughout that
mass.
But the present potential energy of the solar system is so enormous,
approaching in fact possibly to what in our helplessness we call
infinite, that it may supply for absolutely incalculable future ages
what is required for the physical existence of life. Again, the fall
together, from the distance of Sirius let us say, of the sun and an
equal star would at once supply the sun with at least as much energy
for future radiation to possible planets as could possibly have been
acquired by his own materials in their original falling together
from practically infinite diffusion as a cloud of stones or dust, or
a nebula; so that it is certain that, if the present physical laws
remain long enough in operation, there will be (at immense intervals
of time) mighty catastrophes due to the crashing together of
defunct suns—the smashing of the greater part of each into nebulous
dust surrounding the remainder, which will form an intensely heated
nucleus—then, possibly, the formation of a new and larger set of
planets with a proportionately larger and hotter sun, a solar system
on a far grander scale than the present. And so on, growing in
grandeur but diminishing in number till the exhaustion of energy is
complete, and after that eternal rest, so far at least as visible
motion is concerned.[43]
116. The study of the necessary future has prepared us for an inquiry
into the long remote past. Just as the present discrete stellar
systems must finally come together, so the materials which now form
them must have originally been widely separate. Our modern knowledge
enables us to look back with almost certitude to the time when
there was nothing but gravitating matter and its potential energy
throughout the expanse of space—ready, as slight local differences
of distribution predisposed it, to break up into portions, each
converging to one or more nuclei of its own, and thus forming in time
separate solar or stellar systems. We have thus reached the beginning
as well as the end of the present visible universe, and have come to
the conclusion that it began in time and will in time come to an end.
Immortality is therefore impossible in such a universe.