... οἱ οὐρανοὶ ῥοιζηδὸν παρελεύσονται, στοιχεῖα δὲ καυσούμενα λυθήσονται, καὶ γῆ καὶ τὰ ἐν αὐτῇ ἔργα κατακαήσεται.—Πετρού Βʹ. γʹ.
... 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 dieBefore this mortal shall assumeHis immortality.’—Campbell.
‘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 dieBefore this mortal shall assumeHis immortality.’—Campbell.
‘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 dieBefore this mortal shall assumeHis immortality.’—Campbell.
‘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 generallyused 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 ofsubstanceorstuffas 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 whichinvolved either heat or flame, real or apparent), was in early times one of the four so-calledelements.
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 errarecomes 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 thatsomething elsebesides stuff which has anobjectivethough not asubstantive[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 externaluniverse:—
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, dowith 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 theConservation 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 offersabsoluteresistance 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 bein the same senseconserved. (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 conservationof 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 towork 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 suchthingas force (which is suggested to us by the impressions of ourmuscularsense), any more than there is such athingas Sound, or Light, which are mere names for physical impressions producedupon 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’sfirstinterpretation of his Third Law of Motion, viz., thatAction and Reaction are equal and opposite. In thisfirstinterpretation 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 itmomentum, 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 factdirected magnitudes(like the forces of which they are the measure), and are therefore capable ofcancelling one anotherwhentheir 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) anequaland 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, butin the complete systemof 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 destroyedin 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. couplesin 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 vivais 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 essentiallyproducts 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 whatvis vivadepends. It depends upon and is proportional to the product of the mass into thesquareof the velocity. Compare, or rather contrast, this with the definition of momentum given above, and it will be seen thatvis vivais 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 ashalfthe 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 novis vivaof either cannon or ball. After firing each hasvis 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 hasvis vivathough it has no momentum. If,as before, we could reverse the motions of cannon and ball, then, even when they impinged, thevis vivawouldnotbe 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—itsvis 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 asecondinterpretation of his Third Law of Motion. His statement is equivalent, in modern language, to the following:[36]—Work doneon 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 dynamicsrecognise thatit is not lost, but only transformedinto 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 asthe‘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 seriouslacunain 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 frictionin 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 thelacunain 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 movesthrough 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 suchtransmutations 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 onthe only admissible basis—experiment, the grand law of theConservation 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 hisThird Law of Motioninto the modern statement of theConservation 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 muchclaim 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 ofthe sum of the two kindsof 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 orNecessitasof the antients; the other is Proteus himself in the variety and rapidity of its transformations.
Φύσις, διαδόχαις σχημάτωντρισμυρίοιςἀλλάσσεται τύπωμα, Πρωτέως δίκην,πάντων ὅσ’ ἔστι ποικιλώτατον τέρας·τῆς δ’ αὖτ’ Ἀνάγκης ἐστ’ ἀκίνητον σθένος,μόνη δ’ ἁπάντων ταὐτὸ διαμένουσ’ ἀεὶβροτῶν τε καὶ θεῶν πάντ’ ἀποτρύει γένη.[37]
Φύσις, διαδόχαις σχημάτωντρισμυρίοιςἀλλάσσεται τύπωμα, Πρωτέως δίκην,πάντων ὅσ’ ἔστι ποικιλώτατον τέρας·τῆς δ’ αὖτ’ Ἀνάγκης ἐστ’ ἀκίνητον σθένος,μόνη δ’ ἁπάντων ταὐτὸ διαμένουσ’ ἀεὶβροτῶν τε καὶ θεῶν πάντ’ ἀποτρύει γένη.[37]
Φύσις, διαδόχαις σχημάτωντρισμυρίοις
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 lowerpoint 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 thequantityof energy remains for ever unchanged, itsavailabilitysteadily 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,whenall 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 toArt. 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 newideas:—
(1.) That, at least with our present knowledge, no inference is possible as to the relation between heatand work, until the heated or working substance is brought back, after a completeCycleof 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 thefirst to recallattention 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 asfollows:—
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 bemade 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 cannotof itselfpass 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 hegives:—
‘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 guardednote:—
‘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 amplereductio 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 withHeat, 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 unitco-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 theProbability 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 thisthere is—always provided the particles be sufficiently numerous—no perceptible change in the statistics of the group, except in so far as concernsindividualparticles, 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 velocityamong 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 wallsof 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 mereguidanceapplied 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 inArt. 107must 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 ispar excellencethe 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 dueto 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.