Let us next consider a point which belongs to pure mechanics, the history of the principle of virtual motions or virtual velocities. This principle was not first enunciated, as is usually stated, and as Lagrange also a.s.serts, by Galileo, but earlier, by Stevinus. In his Trochleostatica of the above-cited work, page 72, he says: "Observe that this axiom of statics holds good here: "As the s.p.a.ce of the body acting is to the s.p.a.ce of the body acted upon, so is the power of the body acted upon to the power of the body acting."[49]

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

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

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

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

The whole of mechanics, thus, is based upon an idea, which, though unequivocal, is yet unwonted and not coequal with the other principles and axioms of mechanics. Every student of mechanics, at some stage of his progress, feels the uncomfortableness of this state of affairs; every one wishes it removed; but seldom is the difficulty stated in words. Accordingly, the zealous pupil of the science is highly rejoiced when he reads in a master like Poinsot (ThAorie gAnArale de l'Aquilibre et du mouvement des systAmes) the following pa.s.sage, in which that author is giving his opinion of the a.n.a.lytical Mechanics: "In the meantime, because our attention in that work was first wholly engrossed with the consideration of its beautiful development of mechanics, which seemed to spring complete from a single formula, we naturally believed that the science was completed or that it only remained to seek the demonstration of the principle of virtual velocities. But that quest brought back all the difficulties that we had overcome by the principle itself. That law so general, wherein are mingled the vague and unfamiliar ideas of infinitely small movements and of perturbations of equilibrium, only grew obscure upon examination; and the work of Lagrange supplying nothing clearer than the march of a.n.a.lysis, we saw plainly that the clouds had only appeared lifted from the course of mechanics because they had, so to speak, been gathered at the very origin of that science.

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

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

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

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

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

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

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

II. MECHANICAL PHYSICS.

It cannot be denied that an unmistakable tendency has prevailed, from Democritus to the present day, to explain all physical events mechanically. Not to mention earlier obscure expressions of that tendency we read in Huygens the following:[51]

"There can be no doubt that light consists of the motion of a certain substance. For if we examine its production, we find that here on earth it is princ.i.p.ally fire and flame which engender it, both of which contain beyond doubt bodies which are in rapid movement, since they dissolve and destroy many other bodies more solid than they: while if we regard its effects, we see that when light is acc.u.mulated, say by concave mirrors, it has the property of combustion just as fire has, that is to say, it disunites the parts of bodies, which is a.s.suredly a proof of motion, at least in the true philosophy, in which the causes of all natural effects are conceived as mechanical causes. Which in my judgment must be accomplished or all hope of ever understanding physics renounced."[52]

S. Carnot,[53] in introducing the principle of excluded perpetual motion into the theory of heat, makes the following apology: "It will be objected here, perhaps, that a perpetual motion proved impossible for purely mechanical actions, is perhaps not so when the influence of heat or of electricity is employed. But can phenomena of heat or electricity be thought of as due to anything else than to certain motions of bodies, and as such must they not be subject to the general laws of mechanics?"[54]

These examples, which might be multiplied by quotations from recent literature indefinitely, show that a tendency to explain all things mechanically actually exists. This tendency is also intelligible. Mechanical events as simple motions in s.p.a.ce and time best admit of observation and pursuit by the help of our highly organised senses. We reproduce mechanical processes almost without effort in our imagination. Pressure as a circ.u.mstance that produces motion is very familiar to us from daily experience. All changes which the individual personally produces in his environment, or humanity brings about by means of the arts in the world, are effected through the instrumentality of motions. Almost of necessity, therefore, motion appears to us as the most important physical factor. Moreover, mechanical properties may be discovered in all physical events. The sounding bell trembles, the heated body expands, the electrified body attracts other bodies. Why, therefore, should we not attempt to grasp all events under their mechanical aspect, since that is so easily apprehended and most accessible to observation and measurement? In fact, no objection is to be made to the attempt to elucidate the properties of physical events by mechanical a.n.a.logies.

But modern physics has proceeded very far in this direction. The point of view which Wundt represents in his excellent treatise On the Physical Axioms is probably shared by the majority of physicists. The axioms of physics which Wundt sets up are as follows: 1. All natural causes are motional causes.

2. Every motional cause lies outside the object moved.

3. All motional causes act in the direction of the straight line of junction, and so forth.

4. The effect of every cause persists.

5. Every effect involves an equal countereffect.

6. Every effect is equivalent to its cause.

These principles might be studied properly enough as fundamental principles of mechanics. But when they are set up as axioms of physics, their enunciation is simply tantamount to a negation of all events except motion.

According to Wundt, all changes of nature are mere changes of place. All causes are motional causes (page 26). Any discussion of the philosophical grounds on which Wundt supports his theory would lead us deep into the speculations of the Eleatics and the Herbartians. Change of place, Wundt holds, is the only change of a thing in which a thing remains identical with itself. If a thing changed qualitatively, we should be obliged to imagine that something was annihilated and something else created in its place, which is not to be reconciled with our idea of the ident.i.ty of the object observed and of the indestructibility of matter. But we have only to remember that the Eleatics encountered difficulties of exactly the same sort in motion. Can we not also imagine that a thing is destroyed in one place and in another an exactly similar thing created? After all, do we really know more why a body leaves one place and appears in another, than why a cold body grows warm? Granted that we had a perfect knowledge of the mechanical processes of nature, could we and should we, for that reason, put out of the world all other processes that we do not understand? On this principle it would really be the simplest course to deny the existence of the whole world. This is the point at which the Eleatics ultimately arrived, and the school of Herbart stopped little short of the same goal.

Physics treated in this sense supplies us simply with a diagram of the world, in which we do not know reality again. It happens, in fact, to men who give themselves up to this view for many years, that the world of sense from which they start as a province of the greatest familiarity, suddenly becomes, in their eyes, the supreme "world-riddle."

Intelligible as it is, therefore, that the efforts of thinkers have always been bent upon the "reduction of all physical processes to the motions of atoms," it must yet be affirmed that this is a chimerical ideal. This ideal has often played an effective part in popular lectures, but in the workshop of the serious inquirer it has discharged scarcely the least function. What has really been achieved in mechanical physics is either the elucidation of physical processes by more familiar mechanical a.n.a.logies, (for example, the theories of light and of electricity,) or the exact quant.i.tative ascertainment of the connexion of mechanical processes with other physical processes, for example, the results of thermodynamics.

III. THE PRINCIPLE OF ENERGY IN PHYSICS.

We can know only from experience that mechanical processes produce other physical transformations, or vice versa. The attention was first directed to the connexion of mechanical processes, especially the performance of work, with changes of thermal conditions by the invention of the steam-engine, and by its great technical importance. Technical interests and the need of scientific lucidity meeting in the mind of S. Carnot led to the remarkable development from which thermodynamics flowed. It is simply an accident of history that the development in question was not connected with the practical applications of electricity.

In the determination of the maximum quant.i.ty of work that, generally, a heat-machine, or, to take a special case, a steam-engine, can perform with the expenditure of a given amount of heat of combustion, Carnot is guided by mechanical a.n.a.logies. A body can do work on being heated, by expanding under pressure. But to do this the body must receive heat from a hotter body. Heat, therefore, to do work, must pa.s.s from a hotter body to a colder body, just as water must fall from a higher level to a lower level to put a mill-wheel in motion. Differences of temperature, accordingly, represent forces able to do work exactly as do differences of height in heavy bodies. Carnot pictures to himself an ideal process in which no heat flows away unused, that is, without doing work. With a given expenditure of heat, accordingly, this process furnishes the maximum of work. An a.n.a.logue of the process would be a mill-wheel which scooping its water out of a higher level would slowly carry it to a lower level without the loss of a drop. A peculiar property of the process is, that with the expenditure of the same work the water can be raised again exactly to its original level. This property of reversibility is also shared by the process of Carnot. His process also can be reversed by the expenditure of the same amount of work, and the heat again brought back to its original temperature level.

Suppose, now, we had two different reversible processes A, B, such that in A a quant.i.ty of heat, Q, flowing off from the temperature taCA to the lower temperature taCC should perform the work W, but in B under the same circ.u.mstances it should perform a greater quant.i.ty of work W + W'; then, we could join B in the sense a.s.signed and Ain the reverse sense into a single process. Here A would reverse the transformation of heat produced by B and would leave a surplus of work W', produced, so to speak, from nothing. The combination would present a perpetual motion.

With the feeling, now, that it makes little difference whether the mechanical laws are broken directly or indirectly (by processes of heat), and convinced of the existence of a universal law-ruled connexion of nature, Carnot here excludes for the first time from the province of general physics the possibility of a perpetual motion. But it follows, then, that the quant.i.ty of work W, produced by the pa.s.sage of a quant.i.ty of heat Q from a temperature taCA to a temperature taCC, is independent of the nature of the substances as also of the character of the process, so far as that is unaccompanied by loss, but is wholly dependent upon the temperature taCA, taCC.

This important principle has been fully confirmed by the special researches of Carnot himself (1824), of Clapeyron (1834), and of Sir William Thomson (1849), now Lord Kelvin. The principle was reached without any a.s.sumption whatever concerning the nature of heat, simply by the exclusion of a perpetual motion. Carnot, it is true, was an adherent of the theory of Black, according to which the sum-total of the quant.i.ty of heat in the world is constant, but so far as his investigations have been hitherto considered the decision on this point is of no consequence. Carnot's principle led to the most remarkable results. W. Thomson (1848) founded upon it the ingenious idea of an "absolute" scale of temperature. James Thomson (1849) conceived a Carnot process to take place with water freezing under pressure and, therefore, performing work. He discovered, thus, that the freezing point is lowered 00075 Celsius by every additional atmosphere of pressure. This is mentioned merely as an example.

About twenty years after the publication of Carnot's book a further advance was made by J. R. Mayer and J. P. Joule. Mayer, while engaged as a physician in the service of the Dutch, observed, during a process of bleeding in Java, an unusual redness of the venous blood. In agreement with Liebig's theory of animal heat he connected this fact with the diminished loss of heat in warmer climates, and with the diminished expenditure of organic combustibles. The total expenditure of heat of a man at rest must be equal to the total heat of combustion. But since all organic actions, even the mechanical actions, must be set down to the credit of the heat of combustion, some connexion must exist between mechanical work and expenditure of heat.

Joule started from quite similar convictions concerning the galvanic battery. A heat of a.s.sociation equivalent to the consumption of the zinc can be made to appear in the galvanic cell. If a current is set up, a part of this heat appears in the conductor of the current. The interposition of an apparatus for the decomposition of water causes a part of this heat to disappear, which on the burning of the explosive gas formed, is reproduced. If the current runs an electromotor, a portion of the heat again disappears, which, on the consumption of the work by friction, again makes its appearance. Accordingly, both the heat produced and the work produced, appeared to Joule also as connected with the consumption of material. The thought was therefore present, both to Mayer and to Joule, of regarding heat and work as equivalent quant.i.ties, so connected with each other that what is lost in one form universally appears in another. The result of this was a substantial conception of heat and of work, and ultimately a substantial conception of energy. Here every physical change of condition is regarded as energy, the destruction of which generates work or equivalent heat. An electric charge, for example, is energy.

In 1842 Mayer had calculated from the physical constants then universally accepted that by the disappearance of one kilogramme-calorie 365 kilogramme-metres of work could be performed, and vice versa. Joule, on the other hand, by a long series of delicate and varied experiments beginning in 1843 ultimately determined the mechanical equivalent of the kilogramme-calorie, more exactly, as 425 kilogramme-metres.

If we estimate every change of physical condition by the mechanical work which can be performed upon the disappearance of that condition, and call this measure energy, then we can measure all physical changes of condition, no matter how different they may be, with the same common measure, and say: the sum-total of all energy remains constant. This is the form that the principle of excluded perpetual motion received at the hands of Mayer, Joule, Helmholtz, and W. Thomson in its extension to the whole domain of physics.

After it had been proved that heat must disappear if mechanical work was to be done at its expense, Carnot's principle could no longer be regarded as a complete expression of the facts. Its improved form was first given, in 1850, by Clausius, whom Thomson followed in 1851. It runs thus: "If a quant.i.ty of heat Q' is transformed into work in a reversible process, another quant.i.ty of heat Q of the absolute[55] temperature TaCA is lowered to the absolute temperature TaCC." Here Q' is dependent only on Q, TaCA, TaCC, but is independent of the substances used and of the character of the process, so far as that is unaccompanied by loss. Owing to this last fact, it is sufficient to find the relation which obtains for some one well-known physical substance, say a gas, and some definite simple process. The relation found will be the one that holds generally. We get, thus, Q'/(Q' + Q) = (TaCA-TaCC)/TaCA (1) that is, the quotient of the available heat Q' transformed into work divided by the sum of the transformed and transferred heats (the total sum used), the so-called economical coefficient of the process, is, (TaCA-TaCC)/TaCA.

IV. THE CONCEPTIONS OF HEAT.

When a cold body is put in contact with a warm body it is observed that the first body is warmed and that the second body is cooled. We may say that the first body is warmed at the expense of the second body. This suggests the notion of a thing, or heat-substance, which pa.s.ses from the one body to the other. If two ma.s.ses of water m, m', of unequal temperatures, be put together, it will be found, upon the rapid equalisation of the temperatures, that the respective changes of temperatures u and u' are inversely proportional to the ma.s.ses and of opposite signs, so that the algebraical sum of the products is, mu + m'u' = 0.

Black called the products mu, m'u', which are decisive for our knowledge of the process, quant.i.ties of heat. We may form a very clear picture of these products by conceiving them with Black as measures of the quant.i.ties of some substance. But the essential thing is not this picture but the constancy of the sum of these products in simple processes of conduction. If a quant.i.ty of heat disappears at one point, an equally large quant.i.ty will make its appearance at some other point. The retention of this idea leads to the discovery of specific heat. Black, finally, perceives that also something else may appear for a vanished quant.i.ty of heat, namely: the fusion or vaporisation of a definite quant.i.ty of matter. He adheres here still to this favorite view, though with some freedom, and considers the vanished quant.i.ty of heat as still present, but as latent.

The generally accepted notion of a caloric, or heat-stuff, was strongly shaken by the work of Mayer and Joule. If the quant.i.ty of heat can be increased and diminished, people said, heat cannot be a substance, but must be a motion. The subordinate part of this statement has become much more popular than all the rest of the doctrine of energy. But we may convince ourselves that the motional conception of heat is now as unessential as was formerly its conception as a substance. Both ideas were favored or impeded solely by accidental historical circ.u.mstances. It does not follow that heat is not a substance from the fact that a mechanical equivalent exists for quant.i.ty of heat. We will make this clear by the following question which bright students have sometimes put to me. Is there a mechanical equivalent of electricity as there is a mechanical equivalent of heat? Yes, and no. There is no mechanical equivalent of quant.i.ty of electricity as there is an equivalent of quant.i.ty of heat, because the same quant.i.ty of electricity has a very different capacity for work, according to the circ.u.mstances in which it is placed; but there is a mechanical equivalent of electrical energy.

Let us ask another question. Is there a mechanical equivalent of water? No, there is no mechanical equivalent of quant.i.ty of water, but there is a mechanical equivalent of weight of water multiplied by its distance of descent.

When a Leyden jar is discharged and work thereby performed, we do not picture to ourselves that the quant.i.ty of electricity disappears as work is done, but we simply a.s.sume that the electricities come into different positions, equal quant.i.ties of positive and negative electricity being united with one another.

What, now, is the reason of this difference of view in our treatment of heat and of electricity? The reason is purely historical, wholly conventional, and, what is still more important, is wholly indifferent. I may be allowed to establish this a.s.sertion.

In 1785 Coulomb constructed his torsion balance, by which he was enabled to measure the repulsion of electrified bodies. Suppose we have two small b.a.l.l.s, A, B, which over their whole extent are similarly electrified. These two b.a.l.l.s will exert on one another, at a certain distance r of their centres, a certain repulsion p. We bring into contact with B now a ball C, suffer both to be equally electrified, and then measure the repulsion of B from A and of C from A at the same distance r. The sum of these repulsions is again p. Accordingly something has remained constant. If we ascribe this effect to a substance, then we infer naturally its constancy. But the essential point of the exposition is the divisibility of the electric force p and not the simile of substance.

In 1838 Riess constructed his electrical air-thermometer (the thermoelectrometer). This gives a measure of the quant.i.ty of heat produced by the discharge of jars. This quant.i.ty of heat is not proportional to the quant.i.ty of electricity contained in the jar by Coulomb's measure, but if Q be this quant.i.ty and C be the capacity, is proportional to Q/2C, or, more simply still, to the energy of the charged jar. If, now, we discharge the jar completely through the thermometer, we obtain a certain quant.i.ty of heat, W. But if we make the discharge through the thermometer into a second jar, we obtain a quant.i.ty less than W. But we may obtain the remainder by completely discharging both jars through the air-thermometer, when it will again be proportional to the energy of the two jars. On the first, incomplete discharge, accordingly, a part of the electricity's capacity for work was lost.

When the charge of a jar produces heat its energy is changed and its value by Riess's thermometer is decreased. But by Coulomb's measure the quant.i.ty remains unaltered.

Now let us imagine that Riess's thermometer had been invented before Coulomb's torsion balance, which is not a difficult feat, since both inventions are independent of each other; what would be more natural than that the "quant.i.ty" of electricity contained in a jar should be measured by the heat produced in the thermometer? But then, this so-called quant.i.ty of electricity would decrease on the production of heat or on the performance of work, whereas it now remains unchanged; in that case, therefore, electricity would not be a substance but a motion, whereas now it is still a substance. The reason, therefore, why we have other notions of electricity than we have of heat, is purely historical, accidental, and conventional.

This is also the case with other physical things. Water does not disappear when work is done. Why? Because we measure quant.i.ty of water with scales, just as we do electricity. But suppose the capacity of water for work were called quant.i.ty, and had to be measured, therefore, by a mill instead of by scales; then this quant.i.ty also would disappear as it performed the work. It may, now, be easily conceived that many substances are not so easily got at as water. In that case we should be unable to carry out the one kind of measurement with the scales whilst many other modes of measurement would still be left us.

In the case of heat, now, the historically established measure of "quant.i.ty" is accidentally the work-value of the heat. Accordingly, its quant.i.ty disappears when work is done. But that heat is not a substance follows from this as little as does the opposite conclusion that it is a substance. In Black's case the quant.i.ty of heat remains constant because the heat pa.s.ses into no other form of energy.

If any one to-day should still wish to think of heat as a substance, we might allow that person this liberty with little ado. He would only have to a.s.sume that that which we call quant.i.ty of heat was the energy of a substance whose quant.i.ty remained unaltered, but whose energy changed. In point of fact we might much better say, in a.n.a.logy with the other terms of physics, energy of heat, instead of quant.i.ty of heat.

When we wonder, therefore, at the discovery that heat is motion, we wonder at something that was never discovered. It is perfectly indifferent and possesses not the slightest scientific value, whether we think of heat as a substance or not. The fact is, heat behaves in some connexions like a substance, in others not. Heat is latent in steam as oxygen is latent in water.

V. THE CONFORMITY IN THE DEPORTMENT OF THE ENERGIES.

The foregoing reflexions will gain in lucidity from a consideration of the conformity which obtains in the behavior of all energies, a point to which I called attention long ago.[56]

A weight P at a height HaCA represents an energy WaCA = PHaCA. If we suffer the weight to sink to a lower height HaCC, during which work is done, and the work done is employed in the production of living force, heat, or an electric charge, in short, is transformed, then the energy WaCC = PHaCC is still left. The equation subsists WaCA/HaCA = WaCC/HaCC, (2).

or, denoting the transformed energy by W' = WaCA-WaCC and the transferred energy, that transported to the lower level, by W = WaCC, W'/(W' + W) = (HaCA-HaCC)/HaCA, (3).

an equation in all respects a.n.a.logous to equation (1) at page 165. The property in question, therefore, is by no means peculiar to heat. Equation (2) gives the relation between the energy taken from the higher level and that deposited on the lower level (the energy left behind); it says that these energies are proportional to the heights of the levels. An equation a.n.a.logous to equation (2) may be set up for every form of energy; hence the equation which corresponds to equation (3), and so to equation (1), may be regarded as valid for every form. For electricity, for example, HaCA, HaCC signify the potentials.

When we observe for the first time the agreement here indicated in the transformative law of the energies, it appears surprising and unexpected, for we do not perceive at once its reason. But to him who pursues the comparative historical method that reason will not long remain a secret.

Since Galileo, mechanical work, though long under a different name, has been a fundamental concept of mechanics, as also a very important notion in the applied sciences. The transformation of work into living force, and of living force into work, suggests directly the notion of energy--the idea having been first fruitfully employed by Huygens, although Thomas Young first called it by the name of "energy." Let us add to this the constancy of weight (really the constancy of ma.s.s) and we shall see that with respect to mechanical energy it is involved in the very definition of the term that the capacity for work or the potential energy of a weight is proportional to the height of the level at which it is, in the geometrical sense, and that it decreases on the lowering of the weight, on transformation, proportionally to the height of the level. The zero level here is wholly arbitrary. With this, equation (2) is given, from which all the other forms follow.

When we reflect on the tremendous start which mechanics had over the other branches of physics, it is not to be wondered at that the attempt was always made to apply the notions of that science wherever this was possible. Thus the notion of ma.s.s, for example, was imitated by Coulomb in the notion of quant.i.ty of electricity. In the further development of the theory of electricity, the notion of work was likewise immediately introduced in the theory of potential, and heights of electrical level were measured by the work of unit of quant.i.ty raised to that level. But with this the preceding equation with all its consequences is given for electrical energy. The case with the other energies was similar.

Thermal energy, however, appears as a special case. Only by the peculiar experiments mentioned could it be discovered that heat is an energy. But the measure of this energy by Black's quant.i.ty of heat is the outcome of fortuitous circ.u.mstances. In the first place, the accidental slight variability of the capacity for heat c with the temperature, and the accidental slight deviation of the usual thermometrical scales from the scale derived from the tensions of gases, brings it about that the notion "quant.i.ty of heat" can be set up and that the quant.i.ty of heat ct corresponding to a difference of temperature t is nearly proportional to the energy of the heat. It is a quite accidental historical circ.u.mstance that Amontons. .h.i.t upon the idea of measuring temperature by the tension of a gas. It is certain in this that he did not think of the work of the heat.[57] But the numbers standing for temperature, thus, are made proportional to the tensions of gases, that is, to the work done by gases, with otherwise equal changes of volume. It thus happens that temperature heights and level heights of work are proportional to one another.

If properties of the thermal condition varying greatly from the tensions of gases had been chosen, this relation would have a.s.sumed very complicated forms, and the agreement between heat and the other energies above considered would not subsist. It is very instructive to reflect upon this point. A natural law, therefore, is not implied in the conformity of the behavior of the energies, but this conformity is rather conditioned by the uniformity of our modes of conception and is also partly a matter of good fortune.

VI. THE DIFFERENCES OF THE ENERGIES AND THE LIMITS OF THE PRINCIPLE OF ENERGY.

Of every quant.i.ty of heat Q which does work in a reversible process (one unaccompanied by loss) between the absolute temperatures TaCA, TaCC, only the portion (TaCA-TaCC)/TaCA.

is transformed into work, while the remainder is transferred to the lower temperature-level TaCC. This transferred portion can, upon the reversal of the process, with the same expenditure of work, again be brought back to the level TaCA. But if the process is not reversible, then more heat than in the foregoing case flows to the lower level, and the surplus can no longer be brought back to the higher level TaCC without some special expenditure. W. Thomson (1852), accordingly, drew attention to the fact, that in all non-reversible, that is, in all real thermal processes, quant.i.ties of heat are lost for mechanical work, and that accordingly a dissipation or waste of mechanical energy is taking place. In all cases, heat is only partially transformed into work, but frequently work is wholly transformed into heat. Hence, a tendency exists towards a diminution of the mechanical energy and towards an increase of the thermal energy of the world.

For a simple, closed cyclical process, accompanied by no loss, in which the quant.i.ty of heat QaCA is taken from the level TaCA, and the quant.i.ty QaCC is deposited upon the level TaCC, the following relation, agreeably to equation (2), exists, -(QaCA/TaCA) + (QaCC/TaCC) = 0.

Similarly, for any number of compound reversible cycles Clausius finds the algebraical sum [sum]Q/T = 0, and supposing the temperature to change continuously, [integral]dQ/T = 0 (4).

Here the elements of the quant.i.ties of heat deducted from a given level are reckoned negative, and the elements imparted to it, positive. If the process is not reversible, then expression (4), which Clausius calls entropy, increases. In actual practice this is always the case, and Clausius finds himself led to the statement: 1. That the energy of the world remains constant.

2. That the entropy of the world tends toward a maximum.

Once we have noted the above-indicated conformity in the behavior of different energies, the peculiarity of thermal energy here mentioned must strike us. Whence is this peculiarity derived, for, generally every energy pa.s.ses only partly into another form, which is also true of thermal energy? The explanation will be found in the following.

Every transformation of a special kind of energy A is accompanied with a fall of potential of that particular kind of energy, including heat. But whilst for the other kinds of energy a transformation and therefore a loss of energy on the part of the kind sinking in potential is connected with the fall of the potential, with heat the case is different. Heat can suffer a fall of potential without sustaining a loss of energy, at least according to the customary mode of estimation. If a weight sinks, it must create perforce kinetic energy, or heat, or some other form of energy. Also, an electrical charge cannot suffer a fall of potential without loss of energy, i. e., without transformation. But heat can pa.s.s with a fall of temperature to a body of greater capacity and the same thermal energy still be preserved, so long as we regard every quant.i.ty of heat as energy. This it is that gives to heat, besides its property of energy, in many cases the character of a material substance, or quant.i.ty.

If we look at the matter in an unprejudiced light, we must ask if there is any scientific sense or purpose in still considering as energy a quant.i.ty of heat that can no longer be transformed into mechanical work, (for example, the heat of a closed equably warmed material system). The principle of energy certainly plays in this case a wholly superfluous rAle, which is a.s.signed to it only from habit.[58] To maintain the principle of energy in the face of a knowledge of the dissipation or waste of mechanical energy, in the face of the increase of entropy is equivalent almost to the liberty which Black took when he regarded the heat of liquefaction as still present but latent.[59] It is to be remarked further, that the expressions "energy of the world" and "entropy of the world" are slightly permeated with scholasticism. Energy and entropy are metrical notions. What meaning can there be in applying these notions to a case in which they are not applicable, in which their values are not determinable?

If we could really determine the entropy of the world it would represent a true, absolute measure of time. In this way is best seen the utter tautology of a statement that the entropy of the world increases with the time. Time, and the fact that certain changes take place only in a definite sense, are one and the same thing.

VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.

We are now prepared to answer the question, What are the sources of the principle of energy? All knowledge of nature is derived in the last instance from experience. In this sense they are right who look upon the principle of energy as a result of experience.

Experience teaches that the sense-elements [alpha beta gamma delta ...] into which the world may be decomposed, are subject to change. It tells us further, that certain of these elements are connected with other elements, so that they appear and disappear together; or, that the appearance of the elements of one cla.s.s is connected with the disappearance of the elements of the other cla.s.s. We will avoid here the notions of cause and effect because of their obscurity and equivocalness. The result of experience may be expressed as follows: The sensuous elements of the world ([alpha beta gamma delta ...]) show themselves to be interdependent. This interdependence is best represented by some such conception as is in geometry that of the mutual dependence of the sides and angles of a triangle, only much more varied and complex.

As an example, we may take a ma.s.s of gas enclosed in a cylinder and possessed of a definite volume ([alpha]), which we change by a pressure ([beta]) on the piston, at the same time feeling the cylinder with our hand and receiving a sensation of heat ([gamma]). Increase of pressure diminishes the volume and increases the sensation of heat.

The various facts of experience are not in all respects alike. Their common sensuous elements are placed in relief by a process of abstraction and thus impressed upon the memory. In this way the expression is obtained of the features of agreement of extensive groups of facts. The simplest sentence which we can utter is, by the very nature of language, an abstraction of this kind. But account must also be taken of the differences of related facts. Facts may be so nearly related as to contain the same kind of a [alpha beta gamma ...], but the relation be such that the [alpha beta gamma ...] of the one differ from the [alpha beta gamma ...] of the other only by the number of equal parts into which they can be divided. Such being the case, if rules can be given for deducing from one another the numbers which are the measures of these [alpha beta gamma ...], then we possess in such rules the most general expression of a group of facts, as also that expression which corresponds to all its differences. This is the goal of quant.i.tative investigation.

If this goal be reached what we have found is that between the [alpha beta gamma ...] of a group of facts, or better, between the numbers which are their measures, a number of equations exists. The simple fact of change brings it about that the number of these equations must be smaller than the number of the [alpha beta gamma ...]. If the former be smaller by one than the latter, then one portion of the [alpha beta gamma ...] is uniquely determined by the other portion.

The quest of relations of this last kind is the most important function of special experimental research, because we are enabled by it to complete in thought facts that are only partly given. It is self-evident that only experience can ascertain that between the [alpha beta gamma ...] relations exist and of what kind they are. Further, only experience can tell that the relations that exist between the [alpha beta gamma ...] are such that changes of them can be reversed. If this were not the fact all occasion for the enunciation of the principle of energy, as is easily seen, would be wanting. In experience, therefore, is buried the ultimate well-spring of all knowledge of nature, and consequently, in this sense, also the ultimate source of the principle of energy.

But this does not exclude the fact that the principle of energy has also a logical root, as will now be shown. Let us a.s.sume on the basis of experience that one group of sensuous elements [alpha beta gamma ...] determines uniquely another group [lambda mu nu ...]. Experience further teaches that changes of [alpha beta gamma ...] can be reversed. It is then a logical consequence of this observation, that every time that [alpha beta gamma ...] a.s.sume the same values this is also the case with [lambda mu nu ...]. Or, that purely periodical changes of [alpha beta gamma ...] can produce no permanent changes of [lambda mu nu ...]. If the group [lambda mu nu ...] is a mechanical group, then a perpetual motion is excluded.

It will be said that this is a vicious circle, which we will grant. But psychologically, the situation is essentially different, whether I think simply of the unique determination and reversibility of events, or whether I exclude a perpetual motion. The attention takes in the two cases different directions and diffuses light over different sides of the question, which logically of course are necessarily connected.

Surely that firm, logical setting of the thoughts noticeable in the great inquirers, Stevinus, Galileo, and the rest, which, consciously or instinctively, was supported by a fine feeling for the slightest contradictions, has no other purpose than to limit the bounds of thought and so exempt it from the possibility of error. In this, therefore, the logical root of the principle of excluded perpetual motion is given, namely, in that universal conviction which existed even before the development of mechanics and co-operated in that development.